directsignalspaceconstructionluenbergertrackerwith...

Research ArticleDirect Signal Space Construction Luenberger Tracker withQuadratic Least Square Regression for ComputationallyEfficient DoA Tracking

Bukeun Byeon Ho-Kyoung Lee and Do-Sik Yoo

School of Electronic and Electrical Engineering Hongik University Map-Gu Wausan Ro 94 Seoul 04066 Republic of Korea

Correspondence should be addressed to Do-Sik Yoo yoodosikhongikackr

Received 28 August 2019 Accepted 8 May 2020 Published 17 June 2020

Academic Editor Ludovic Chamoin

Copyright copy 2020 Bukeun Byeon et alis 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

In this paper we propose a computationally efficient direction-of-arrival (DoA) tracking scheme called the direct signal spaceconstruction Luenberger tracker (DSPCLT) with quadratic least square (QLS) regression Also we study analytically how thechoice of observer gain affects the algorithm performance and illustrate how we can use the theoretical results in determiningoptimal observer gain value e proposed scheme (DSPCLT) has several distinct features compared with existing algorithmsFirst it requires only a fraction of computational complexity compared with other schemes Secondly it maintains robustness bytreating separately the special case of object overlap in which subspace-based algorithms often suffer from lack of resolvabilityirdly the proposed scheme achieves enhanced performance by a method of delay compensation which accounts for ob-servation delay rough numerical analysis we show that DSPCLT achieves performance similar or superior to existing al-gorithms with only a fraction of computational requirement

1 Introduction

In this paper we propose a computationally efficient di-rection-of-arrival (DoA) tracking scheme called the directsignal space construction Luenberger tracker (DSPCLT)with quadratic least square (QLS) regression During the lastseveral decades various subspace-based schemes have beenproposed for DoA estimation such as multiple signal clas-sification algorithms [1] estimation of signal parameter viarotational invariance technique (ESPRIT) [2] and root-MUSIC method [3] Recently because of their high reso-lution these algorithms have been studied in various ap-plications such as nonlinear arrays [4ndash6] 2-dimensionalarrays [7 8] andMIMO radar [9] However early subspace-based algorithms generally suffer from the issue of largecomputational complexity typically due to the need foreigenvalue decomposition (ED) or singular-value decom-position (SVD) For this reason various schemes have beenproposed to reduce the system complexity such as thepropagator method (PM) [10] orthogonal propagator

method (OPM) [11] subspace method without eigende-composition (SWEDE) [12] direct signal space constructionmethod (DSPCM) [13ndash15] and projection approximationsubspace tracking-based deflation (PASTd) algorithm [16]

However these algorithms exhibit difficulty in esti-mating the DoAs when one or more signals suddenly dis-appear or when DoAs overlap [17] As means to overcomesuch issues a number of DoA tracking schemes have beenproposed in [18ndash20] Recently authors in [21 22] proposedto combine existing tracking techniques with computationalefficient subspace techniques to avoid the issues of over-lapping or suddenly disappearing targets while keeping thecomplexity minimal For example the authors in [21]proposed an algorithm by combining PASTd and Kalmanfiltering [23] We shall refer to this algorithm as PASTdKalman in this paper PASTd Kalman reduces the compu-tational complexity of constructing the subspace by avoidingED and SVD with recursive least square (RLS) techniqueHowever the algorithm still requires relatively high com-plexity due to the adaptation of Kalman filtering More

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 4697381 17 pageshttpsdoiorg10115520204697381

recently the authors in [22] proposed an algorithm calledthe adaptive method of estimating DoA (AMEND) whichreplaces the Kalman filter with Luenberger state observer[24] However the complexity of AMEND is not smallerthan PASTd Kalman because it employs for subspaceconstruction a relatively computationally expensive algo-rithm called the subspace-based method without eigende-composition (SUMWE) [25]

In this paper we propose DSPCLT that achieves similaror better performance with computational complexitysignificantly lower than existing algorithms e proposedscheme has several distinct features compared with existingschemes First it achieves very low complexity byemploying DSPCM [14 15] for subspace construction andLuenberger observer in place of the Kalman filter for es-timation and filtering Secondly it achieves robustness bytreating separately the special case of object overlap inwhich subspace-based algorithms suffer from the lack ofresolvability irdly the proposed scheme achieves en-hanced performance by a method of delay compensationwhich takes into account the fact that the DoA estimationat present time is performed with observed data collectedpreviously We show through numerical analysis that theproposed scheme achieves performance similar or superiorto existing algorithms with only a fraction of computationalrequirement

Another important topic studied in this paper is thechoice of observer gains used to implement Luenbergerobserver In most existing schemes employing Luenbergerobserver optimal observer gain values are determinedthrough simulations For example a certain observer gainvalue was suggested in [22] after simulations However wenote that there is infinite variety of situations depending onthe number of antenna sensor elements the numbers ofobjects the speeds of movements the signal-to-noise ratiosof the target the shape of object trajectory and so on Hencefor a selection of a certain observer gain value to be acceptedas a valid choice there must be some degree of guaranteethat a particular observer gain value will provide optimal ornear-optimal performance in other situations which isgenerally very difficult with pure simulations For thisreason we study analytically how the choice of observer gainaffects the algorithm performance and how we can chooseoptimal observer gain value

is paper is organized as follows In Section 2 thesystem model and basic assumptions are described to-gether with the notational definitions used throughoutthis paper In Sections 3 sim 5 the proposed scheme(DSPCLT) is delineated which consists of three stagescalled initialization prediction and estimatione stagesof initialization and prediction are introduced in Section 3and that of estimation is presented in Sections 4 and 5depending on whether objects are resolvable by thesubspace-based algorithm employed In Section 6 westudy theoretically how the selection of observer gainvalue affects the algorithm performance and describe howwe can make optimal observer gain value selection usingthe analytical results Next we show in Section 7 that theproposed scheme achieves similar or better performance

with only a fraction of computational requirementscompared with other algorithms Finally we draw con-clusion in Section 8

e following notations are used throughout this paperBold-faced letters are used for matrices and vectors ecomplex conjugation the transposition and the Hermitiantransposition of matrix A are denoted respectively by AlowastAT and AH e M times N zero matrix and N times N identitymatrix are denoted respectively by OMtimesN and IN

2 System Model and Basic Assumptions

We consider a uniform linear array (ULA) consisting of N

sensors with intersensor spacing d and assume that K

narrowband far-field signal waves s1(t) s2(t) middot middot middot sK(t) ofwavelength λ impinge on the array at distinct directionsrespectively making angles θ1(t) θ2(t) middot middot middot θK(t) with thelinear array We denote by x1(t) x2(t) middot middot middot xN(t) the re-ceived noisy signals collected at the N sensors en if wedefine the steering vector a(θk(t)) by a(θk(t))

[1 ej]k(t) middot middot middot ej(Nminus 1)]k(t)]T with ]k(t) 2πd cos θk(t)λ thesystem can be modeled with the following equation

x(t) A θ1(t) middot middot middot θK(t)( 1113857s(t) + n(t) (1)

where x(t) [x1(t) x2(t) middot middot middot xN(t)]T A(θ1(t) middot middot middot

θK(t)) [a(θ1(t)) middot middot middot a(θK(t))] s(t) [s1(t) middot middot middot

sK(t)]T and n(t) is the vector consisting of the noises addedat the sensors In this paper we make on the system modelthe following basic assumptions

(1) e signal wavelength λ is known a priori and that d

is chosen to be λ2 And the number K of impingingsignals is predetermined by appropriate methodssuch as the one in [26]

(2) e vectors s(t) and n(t) are statistically indepen-dent complex random vector processes with zeromean satisfying

E s(t)s(s)H

1113960 1113961 Pδts

E s(t)s(s)T

1113960 1113961 OKtimesK

E n(t)n(s)H

1113960 1113961 N0δtsIN

E n(t)n(s)T

1113960 1113961 ONtimesN

(2)

where δts is the Kronecker delta function We furtherassume that P is a diagonal matrix with positive realnumbers p1 middot middot middot pK on its diagonal and that thevalue of N0 is assumed to be estimated accurately bythe receiver through long-term observation of thenoise

(3) e system tracks the DoAs by obtaining angle es-timates at time t 0 T 2T 3T where T is apredefined positive real number We assume furtherthat T is chosen to be small enough that the DoAs areessentially constant during the time duration [(n minus

1)T nT] for each n 1 2 3

2 Mathematical Problems in Engineering

3 Initialization and Prediction

In this paper we propose a computationally efficient DoAtracking scheme called the direct signal space constructionLuenberger tracker (DSPCLT) with quadratic least squareregression As in AMEND the DoA is tracked by sequentialupdate of state that is defined as a three-dimensional vectorconsisting of direction-of-arrival and its first and secondderivatives e process of update is carried out in twostages which we call prediction and estimation In theprediction stage the state is tentatively updated purely basedon the previous state en more accurate updated state isobtained in the estimation stage which is the major part ofthe proposed scheme

As in AMEND we propose to use in the estimationstage the Luenberger observer using a subspace-based al-gorithm In particular we use Newton iteration method toobtain the improved estimate of the DoA for use in theLuenberger observer While such subspace-based trackingschemes provide excellent performance in general we needto take special care when the DoAs of objects overlap inwhich case the Newton iteration may not provide robustestimate of the DoA Consequently it is wise to considersuch a special case separately

Since it is rather complicated to describe the estimationstage we describe only the initialization stage and the pre-diction stage in this section en in the next two sections weconsider the estimation stage in nonresolvable and resolvablesituations respectively e proposed scheme employs foreach object a 3-dimensional state vector whose componentsrepresent the direction-of-arrival and its first and second de-rivatives respectively At the initialization stage the state vectoris assumed to be obtained by a certain existing method such asMUSIC [1] After the initialization stage the state vector isupdated at a regular interval T in two stages called predictionand estimation described in the following

Before proceeding let us clarify the notations used in thefollowing First we denote by 1113954yk[n | n] the state vector attime t nT and by 1113954θk[n | n] the first component of 1113954yk[n | n]We shall regard 1113954θk[n | n] as the estimate of the DoA θk(nT)

of object k at time nT Next we denote by 1113954yk[n | n minus 1] thepredicted state vector obtained in the prediction stage attime t nT and by 1113954θk[n | n minus 1] the first component of1113954yk[n | n minus 1]

31 Initialization Stage e initialization stage begins byobtaining the estimates of θk(minus 2T) θk(minus T) and θk(0) fork 1 2 middot middot middot K through a certain DoAmeasurement schemesuch as MUSIC [1] ESPRIT [2] or DSPCM [14 15] eestimates θk(minus 2T) θk(minus T) and θk(0) shall be denotedrespectively by 1113957θk[minus 2 | minus 2] 1113957θk[minus 1 | minus 1] and 1113957θk[0 | 0]en the initial state vector 1113954yk[0 | 0] is computed by

1113954yk[0 | 0] 1113957θk[0 | 0] 1113957θk[0 | 0] minus 1113954θk[minus 1 | minus 1]1113872 1113873T 1113957θk[0 | 0]11138721113960

minus 21113957θk[minus 1 | minus 1] + 1113957θk[minus 2 | minus 2])T2]T (3)

We note that the third component of 1113954yk[0 | 0] is set to bezero in AMEND [22] and PASTd Kalman [21] In fact this

initialization provides better performance in the proposedscheme as in AMEND and PASTd Kalman according to oursimulations Consequently we shall use this method ofinitialization previously used in AMEND and PASTd Kal-man when we evaluate the performance in a later sectionHowever we shall use initialization (3) for the proposedscheme despite slightly worse performance is is becausemathematical analysis in later sections is slightly simplerwith (3)

32PredictionStage After initialization the state vectors areupdated periodically with period T in two stages calledprediction and estimation We assume that the state vector1113954yk[n minus 1 | n minus 1] at previous time (n minus 1)T is given wheren isin 1 2 middot middot middot We assume that 1113954yk[n minus 1 | n minus 1] has beenobtained in the initialization stage if n 1 and in the pre-vious estimation stage if ngt 1 In this stage namely in theprediction stage rough estimates of the next state which wedenote by 1113954yk[n | n minus 1] are obtained by the following dy-namic matrix equation

1113954yk[n | n minus 1] F middot 1113954yk[n minus 1 | n minus 1] (4)

where

F

1 T 05T2

0 1 T

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (5)

Here the first second and third components of thevector 1113954yk[n | n minus 1] are interpreted to represent predictedvalues based on the previous state vector for θk(nT)_θk(nT) and euroθk(nT) respectively After the prediction stagethe distance

dkl 1113954θk[n | n minus 1] minus 1113954θl[n | n minus 1]11138681113868111386811138681113868

11138681113868111386811138681113868 (6)

is computed for l 1 middot middot middot k minus 1 k + 1 middot middot middot K e object k isregarded to be in the nonresolvable situation if dkl is lessthan or equal to a certain predetermined threshold value forsome i j with lne k Naturally the object k is regarded to beresolvable if it is not in the nonresolvable situation

4 Estimation Stage in theNonresolvable Situation

We note that the nonresolvable situation becomes less likelyif the number of antenna elements is increased In otherwords the nonresolvable situation does not happen veryoften and does not last very long even when it happens if thenumber of antenna elements is large enough Although thenonresolvable situation may not happen very often and thealgorithm used in this situation is not themajor contributionof this paper we consider this situation first because theproposed algorithm for the resolvable situation is muchmore complicated to describe

In the nonresolvable situation the estimated state vector1113954yk[n | n] is obtained by using quadratic least square regressionusing the past p estimated DoAs 1113954θk[m | m] m n minus p nminus

Mathematical Problems in Engineering 3

p + 1 middot middot middot n minus 1 To be more specific we assume that the trueDoA θ[m] satisfies

θk[m] a(m minus n)2

+ b(m minus n) + c (7)

and attempt to obtain the constants a b and c by the leastsquare method with the previously estimated DoAs1113954θk[m | m] m n minus p n minus p + 1 middot middot middot n minus 1 We note that theresultant constants a b and c are given by [27]

a

b

c

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p 1113936minus 1

mminus p

m 1113936minus 1

mminus p

m2

1113936minus 1

mminus p

m 1113936minus 1

mminus p

m2 1113936minus 1

mminus p

m3

1113936minus 1

mminus p

m2 1113936minus 1

mminus p

m3 1113936minus 1

mminus p

m4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 1

middot

1113936nminus 1

mnminus p

1113954θk[m | m]

1113936nminus 1

mnminus p

(m minus n)θk[m | m]

1113936nminus 1

mnminus p

(m minus n)21113954θk[m | m]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(8)

Once the constants a b and c are computed by (8) theestimated state vector 1113954yk[n | n] is obtained by

1113954yk[n | n]

c

b

2a

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (9)

We note that the second and third components of1113954yk[n | n] can be obtained by the first and the second de-rivative of θk[m] a(m minus n)2 + b(m minus n) + c at m n

5 Estimation Stage in the Resolvable Situation

In this section we describe howwe obtain the estimated statevector in the resolvable situation First we recall that thepredicted state vector 1113954yk[n | n minus 1] is obtained in the pre-diction stage by simply multiplying the previous estimatedstate vector 1113954yk[n minus 1 | n minus 1] by a constant matrix F withoutreflecting the data collected between times (n minus 1)T and nTWhile the data collected captured at the antenna are stillnot used to obtain the updated estimated state vector in thenonresolvable situation the data are used to obtain theupdated estimated state vector in the resolvable situation Inthe following we describe how the data are used to obtain1113954yk[n | n] in the resolvable situation

In particular the updated estimated state vector 1113954yk[n | n]

is obtained using the Luenberger observer by

1113954yk[n | n] 1113954yk[n | n minus 1] + 1113957θk[n | n] minus 1113954θk[n | n minus 1]1113872 1113873gk (10)

where gk denotes a 3-dimensional column vector calledobserver gain [24] Here the quantity 1113957θk[n | n] is an estimateof θk(nT) obtained by a certain algorithm using the datacollected at the antenna In the following we describe how

we obtain 1113957θk[n | n] in the proposed scheme Before pro-ceeding we note that the performance of the system candepend on the choice of the observer gain gk In many cases asin AMEND no serious discussion is provided regarding howthe choice of the observer gain affects the performance or howwe can achieve optimal or near-optimal performance Wediscuss this issue in the next section Hence we focus in thissection solely on how we obtain the quantity 1113957θk[n | n] whichwe call the delay compensated direct signal space constructionmethod Newton estimate (DCDSPCMNE) of θk(nT) eDCDSPCMNE 1113957θk[n | n] is obtained in two steps In the firststep we obtain a quantity θk[n | n] called the direct signalspace construction method Newton estimate (DSPCMNE)en in the second step we obtain more accurate estimate1113957θk[n | n] by a method of motion compensation

51 Direct Signal Space Construction Method NewtonEstimate Let us first discuss how we obtain DSPCMNEθk[n | n] In many subspace-based DoA estimation schemesthe DoAs are estimated by minimizing a certain costfunction f(θ) defined by

f(θ) aH(θ)Πa(θ) (11)

where Π is the orthogonal projector into the noise subspaceOne major source of computational complexity in subspace-based schemes is in the construction of the orthogonalprojectorΠ In this paper we propose to use the direct signalspace construction method [14 15] to reduce the compu-tational complexity significantly In Appendix A1 it wasdescribed how we can construct the orthogonal projector Πin the proposed scheme

Since the minimum search process also takes somenonnegligible computational complexity various methodshave been proposed to reduce the burden One suchmethodwhich is particularly useful when some coarse estimate of theDoA is available is to employ Newtonrsquos iteration method ofsearching zeroes with fprime(θ) [11 12 22 25] We also adoptNewtonrsquos iteration method in this paper Let θ denote thecoarse estimate of a zero of fprime(θ) en according toNewtonrsquos method a refined zero 1113957θ is given by

1113957θ θ minusfprime(θ)

fPrime(θ) θ minus

Re aprimeH(θ)Πa(θ)1113876 1113877

Re aPrimeH(θ)Πa(θ)1113960 1113961 + aprimeH(θ)Πaprime(θ)

(12)

where Re[middot] denotes the real part of middot Here we note that thefirst term of the denominator is much smaller than thesecond term of the denominator since Πa(θ) asymp 0 For thisreason the first term of the denominator is open dropped inimplementations

Using (12) and neglecting the first term of the de-nominator we obtain DSPCMNE θk[n | n] by

θk[n | n] 1113954θk[n | n minus 1] minusRe dH[n] 1113954Π[n]a[n]1113960 1113961

dH[n] 1113954Π[n]d[n] (13)

where ak[n] and dk[n] are vectors defined respectively by


ak[n] a 1113954θk[n | n minus 1]1113872 1113873 (14)

and

dk[n] da(θ)

dθ

111386811138681113868111386811138681113868 θ1113954θk[n | nminus 1] (15)

Also the matrix 1113954Π[n] in (13) denotes an estimate of thepropagator π We discuss in Appendix A2 how we canconstruct the estimate 1113954Π[n] in the proposed scheme

52 Delay Compensated Direct Signal Space ConstructionMethod Newton Estimate We note that the orthogonalprojector 1113954Π[n] is obtained by using the data collected during thetime interval ((n minus 1)T nT] Hence while DSPCMNE θk[n | n]

is an estimate of θk(t) it is not exactly at time t nT In thissection we regard θk[n | n] as an estimate of θk(t) at a certaintime t (n minus α)T (0lt αlt 1) and obtain the DCDSPCMNE1113957θk[n | n] by compensating the movement during the delay αTFor this purpose we first need to assess the value of α and thenestimate the amount of change in θk(t) during this delay

Before proceeding we shall assume that T and euroθk(t) aresmall enough so that _θk(t) does not change appreciablyover time period [(n minus 1)T nT] However we note evenwith this simplifying assumption that it is very difficult todetermine theoretically justifiable value for αT since theprocess of obtaining 1113954Π[n] does not depend linearly on theobservation times As described in Appendix A2 the dataare collected at times t [n minus (Ns minus 1)δ]T [n minus (Ns minus 2)δ]

T middot middot middot [nminus (Ns minus Ns)]T and forgetting factor ρ is used togive more weight to more recent data In the special case inwhich δ 1Ns and ρ 1 we note that the data are collectedat times t (n minus 1 + 1Ns)T (n minus 1 + 2Ns)T middot middot middot nT andall the data are equally weighted since ρ 1 Consequentlywe observe that the data are collected symmetrically aroundtime t (n minus (Ns minus 1)2Ns)T and are equally used While itis theoretically difficult to justify the validity we shall assessthe value of α as (Ns minus 1)2Ns in this special case

e assessment of the value of α is more difficult in thegeneral case In the general case considering the fact that thedata are collected at t [n minus (Ns minus 1)δ]T [n minus (Nsminus

2)δ]T middot middot middot [n minus (Ns minus Ns)]T and that data collected at timet (n minus (Ns minus l)δ)T are weighted by (1 minus ρ)ρNsminus l(1 minus ρNs )

in the computation of 1113957rm in (A3) and (A4) we shall regardthe weighted time average

n minus Ns minus 1( 1113857δ1113858 1113859T middot(1 minus ρ)ρNsminus 1

1 minus ρNs

+ n minus Ns minus 2( 1113857δ1113858 1113859T middot(1 minus ρ)ρNsminus 2

1 minus ρNs

+ middot middot middot + n minus Ns minus Ns( 1113857δ1113858 1113859T middot(1 minus ρ)ρNsminus Ns

1 minus ρNs

n minusNs minus 1( 1113857ρNs+1 minus NsρNs + ρ

(1 minus ρ) 1 minus ρNs( 1113857middot δ1113890 1113891T

(16)

as the estimated observation time (n minus α)TNext we note that

θk[n | n] minus 1113954θk[n minus 1 | n minus 1] (17)

accounts for the angle change between times (n minus 1)T and(n minus α)T e change between times (n minus 1)T and nT can beestimated as

11 minus α

θk[n | n] minus 1113954θk[n minus 1 | n minus 1]1113966 1113967 (18)

assuming that angular velocity does not change over[(n minus 1)T nT] Using this approximation we obtain theDCDSPCMNE 1113957θk[n | n] by the following equation

1113957θk[n | n] 1113954θk[n minus 1 | n minus 1] +1

1 minus αθk[n | n] minus 1113954θk[n minus 1 | n minus 1]1113966 1113967

(19)

1113954θk[n minus 1 | n minus 1]

+(1 minus ρ) 1 minus ρNs( 1113857 θk[n | n] minus 1113954θk[n minus 1 | n minus 1]1113966 1113967

(1 minus ρ) 1 minus ρNs( 1113857 minus Ns minus 1( 1113857ρNs+1 minus NsρNs + ρ1113864 1113865δ

(20)

Before proceeding we note that (20) reduces to

1113957θk[n | n] 1113954θk[n minus 1 | n minus 1] +NS(1 minus ρ) 1 minus ρNs( 1113857

Ns minus Ns + 1( 1113857ρ + ρNs+11113858 1113859

middot θk[n | n] minus 1113954θk[n minus 1 | n minus 1]1113966 1113967

(21)

in the special case of δ 1Ns

6 Selection of Observer Gain

In Section 5 we discussed how the updated estimated statevector is obtained in the normal situation namely in theresolvable situation In particular the estimated state vector isupdated in the proposed scheme using the Luenbergerobserver as described in (10) One important issue that wasnot discussed in Section 5 was the selection of the observergain gk If certain conditions are met the estimation errorsapproach zero as time progresses For example the state errorvectors converge to zero if and only if all the eigenvalues of thematrix F minus gk 1 0 01113858 1113859 have magnitudes strictly less than onein the case of the algorithm in [22] However not so muchinformation is available about the effect of observer gainselection beyond such theoretical basics Hence observergains are generally selected through simulations Howeverthe question arises whether an observer gain selected to beoptimal through simulations in certain situations will lead toreasonably good performance in other situations In thissection we study the effect of observer gain selection throughtheoretical and numerical analysis and describe how we canobtain reasonably optimal observer gains

61 State Estimation Error We first define the state esti-mation error ek[n] by

ek[n] 1113954yk[n | n] minus yk(nT) (22)


where yk(t) denotes the state vector defined as[θk(t) _θk(t) euroθk(t)]T Also we define a vector δk[n] whichwe call state evolution mismatch at time t nT by

δk[n] yk(nT) minus F middot yk((n minus 1)T) (23)

As discussed in [22] this vector δk[n] equals to a zerovector if the angular acceleration does not change betweentimes (n minus 1)T and nT However the angular accelerationmay change although the amount may be small and δk[n]

becomes small but nonzero vector We define the anglemeasurement error εk[n] by

εk[n] 1113957θk[n | n] minus θk(nT) (24)

In the next proposition we obtain recurrence relationbetween ek[n] and ek[n minus 1]

Proposition 1 9e state estimation errors satisfy the fol-lowing recurrent relationship

ek[n] F minus gkcTF1113872 1113873 middot ek[n minus 1] + gkεk[n] + cTδk[n]gk minus δk[n]

(25)

Proof We first note from (10) (4) and (24) that

1113954yk[n | n] F1113954y[n minus 1 | n minus 1] + θk(nT) + εk[n] minus 1113954θk[n | n minus 1]1113966 1113967gk

(26)

Next from (22) (23) and (26) it follows that

ek[n] F1113954y[n minus 1 | n minus 1] + θk(nT) + εk[n] minus 1113954θk[n | n minus 1]1113966 1113967gkminus F middot yk((n minus 1)T) + δk[n]1113864 1113865

F 1113954y[n minus 1 | n minus 1] minus yk((n minus 1)T)1113864 1113865

+ θk(nT) + εk[n] minus 1113954θk[n | n minus 1]1113966 1113967gk minus δk[n]

(27)

Now since θk(t) cTyk(t) and 1113954θk[n | n minus 1] cT1113954yk[n |

n minus 1] we next obtain

ek[n] F middot ek[n minus 1] + cT yk(nT) minus 1113954yk[n | n minus 1]1113864 1113865gk

+ εk[n]gk minus δk[n](28)

F middot ek[n minus 1] + cT F middot yk((n minus 1)T) + δk[n]1113864

minus F middot yk[n minus 1 | n minus 1]1113865gk + εk[n]gk minus δk[n](29)

Finally we obtain (25) by rearranging (29)

62 Root Mean Squared Error and Observer Gain In thissection we attempt to study theoretically how the observergain affects the performance of angle estimation In par-ticular we study how the selection of observer gain is relatedto the root mean squared error (RMSE) of the DoA of the kth

object which is defined by

RMSEk

1NT

1113944

NT

n1E 1113954θk[n | n] minus θk(nT)

11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(30)

where NT is the total number of observation intervalsBefore proceeding we note that RMSEk can be expressedusing ek[n] as

RMSEk

1NT

1113944

NT

n1E cTek[n]

11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(31)

since1113954θk[n | n] minus θk(nT) cT

1113954yk[n | n] minus cTy(nT)

cTek[n](32)

First we diagonalize the matrix F minus gkcTF into VΛVminus 1where Λ is a diagonal matrix whose diagonal elements areeigenvalues of F minus gkcTF and V is a matrix whose columnvectors are the corresponding eigenvectors Here we notethat it is possible to represent gk in terms of Λ and V asdescribed in the following lemma

Lemma 1

gk V[I minus Λ]Vminus 1c (33)

Proof Since F minus gkcTF VΛVminus 1 it follows that

gkcTFc Fc minus VΛVminus 1c (34)

Here we note that cTFc is simply the (1 1) componentof the matrix F which is 1 Consequently we obtain

gk Fc minus VΛVminus 1c

c minus VΛVminus 1c(35)

from which (33) readily follows

Moreover the observer gain vector is determined solelyby the eigenvalues namely the diagonal elements of Λ asdescribed in the following lemma

Lemma 2 Let κ1 κ2ejϕ and κ2eminus jϕ denote the diagonalelements of Λ where κ1 and κ2 are nonnegative real numbersand ϕ is a real number (Since the characteristic equation ofF minus gkcTF is a third-order equation with real coefficients atleast one of the three eigenvalues must be real-valued and theother two must form a complex conjugate pair) 9en

gk1 1 minus κ1κ22

gk2 12T

3κ1κ22 minus κ22 minus 2κ1κ2 cos ϕ minus κ1 minus 2κ2 cos ϕ + 31113960 1113961

gk3 1

T2 minus κ1κ22 + κ22 + 2κ1κ2 cos ϕ minus κ1 minus 2κ2 cos ϕ + 11113960 1113961

(36)

where gki denotes the ith element of gk

Proof We note that the characteristic equation of F minus gkcTFis given by


λ3 + gk1 + gk2T + 05gk3T2

minus 31113872 1113873λ2

+ minus 2gk1 minus gk2T + 05gk3T2

+ 31113872 1113873λ + gk1 minus 1 0(37)

Consequently for κ1 κ2ejϕ and κ2eminus jϕ to be solutions to(37) the identity equation

λ3 + gk1 + gk2T + 05gk3T2

minus 31113872 1113873λ2


+ 31113872 1113873λ + gk1 minus 1

λ3 minus κ1 + 2κ2 cosϕ( 1113857κ2 + κ2 κ2 + 2κ1 cosϕ( 1113857λ minus κ1κ22 0

(38)

must hold Now comparing the coefficients of the left andright-hand sides of (38) and solving for gki we obtain theresults

Although the result in Lemma 1 is simple it is importantin that we can construct the observer gain vector gk bysuitably choosing the eigenvectors and eigenvalues thatdirectly affect the system performance Before proceedingwith the performance analysis let us rewrite (25) as

ek[n] VΛVminus 1ek[n minus 1] + V(I minus Λ)Vminus 1cεk[n]

+ V(I minus Λ)Vminus 1ccTminus I1113960 1113961δk[n]

(39)

by using (33) By repeatedly using (39) we can represent thestate estimation error ek[n] in terms of its initial value ek[0]the angle measurement error εk[m] and the state evolutionmismatch δk[m] as described in the following

ek[n] VΛnVminus 1ek[0] + 1113944n

m1VΛnminus m

(I minus Λ)Vminus 1cεk[m]

+ 1113944n

m1VΛnminus m

(I minus Λ)Vminus 1ccTminus Vminus 1

1113960 1113961δk[m]

(40)

We note that the second and third terms of (38) involvesummation from m 1 to m n Consequently the secondand third terms may grow larger as n grows However if wechoose the magnitudes of all the diagonal elements of Λ tobe sufficiently less than one only the error εk[m] andmismatch δk[m] with m close to n affect the values of ek[n]

since Λk converges to a zero matrix as k grows Moreoverwe note that the magnitude of δk[m] is order of magnitudessmaller than that of εk[n] unless the observation time in-terval T is chosen to be very large in which the trackingscheme in this paper does not work properly Conse-quently if we choose the magnitudes of all the diagonalelements of Λ to be small enough compared with 1 thethird term of (40) is negligibly small in usual situationsConsequently in the following we shall assume that ek[n]

can be approximated as


m1VΛnminus m


(41)

and attempt to compute RMSEk using (31) To obtainreasonably simple and useful result for RMSEk we shallassume that the angle measurement error εk[m] has zeromean and that εk[m] and εk[n] are uncorrelated if nnemwhich are reasonably accurate assumptions in real situa-tions In the following theorem we summarize the result

Theorem 1 Assume that

E εk[n]1113858 1113859 0 (42)

and that

E εk[m]εk[n]1113858 1113859 δmnσ2k[n] (43)

en

RMSEk

1NT

1113944

NT

n1cTVΛnVminus 1T VΛnVminus 1

1113872 1113873Tc + 1113944

n

m1cTVΛnminus m

(I minus Λ)Vminus 1c1113872 11138732σ2k[m]⎡⎣ ⎤⎦

11139741113972

(44)

where

T

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T

σ2k[0] + σ2k[minus 1]

T2σ2k[0] + 2σ2k[minus 1]

T3

σ2k[0]


T3σ2k[0] + 4σ2k[minus 1] + σ2k[minus 2]

T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(45)


Proof See Appendix B

Corollary 1 If we further assume in 9eorem 1 that thereexists some nonnegative real number 1113957σk such that

σk[n] 1113957σk (46)

for all n then

RMSEk 1113957σk

1NT

1113944

NT


1113872 1113873Tc + 1113944

n

m1cTVΛnminus m

(I minus Λ)Vminus 1c1113872 11138732

⎡⎣ ⎤⎦

11139741113972

(47)

where

T

11T

1T2

1T

2T2

3T3

1T2

3T3

6T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(48)

According to our test with various simulation envi-ronments assumptions (42) and (43) hold very accuratelyHowever assumption (46) does not hold and the value ofσ2k[n] usually fluctuates between 12 and 2 times of theaverage value Despite the invalidity of the assumption theresult in Corollary 1 is very useful First we note the sim-plicity of (47) compared with (44) Consequently it is mucheasier with (47) to grasp intuition about the behavior ofRMSEk in relation to the choice of Λ In particular we notethat there is no further information required about thevariation of σ2k[n] over n and the RMSEk is represented inthe unit of 1113957σk in (47) Secondly we can regard (47) with1113957σk

(1NT) 1113936NT

n σ2k[n]

1113969

as a simple and reasonable ap-proximation of (44) since there is no reason to believe thatσ2k[n] will fluctuate extremely over n

Now we shall show how we can select eigenvalues foroptimal or near-optimal performance with (47) which is notenvironment dependent Later in this section we shall alsodiscuss how much the results with (47) are different fromthose with (44) using a number of simulation environmentsetups We first choose eigenvalues κ1 κ2ejϕ and κ2eminus jϕ andcompute the right-hand side of (47) Apparently we need toobtain V for this purpose However we can computeF minus gkcTF which equals to VΛVminus 1 using Lemma 2 Wefurther note that the right-hand side of (44) can be evaluatedonce we obtain VΛVminus 1

We recall that ek[n] converges to zero as n grows if andonly if both κ1 and κ2 are smaller than 1 We consider005 010 middot middot middot 095 as candidate values for κ1 and κ2 and0deg 5deg middot middot middot 175deg as the values for ϕ Consequently we con-sider a total of 19 times 19 times 36 eigenvalue set combinationsOut of all considered values the case of κ1 07 κ2 05and ϕ 35deg provides the lowest RMSEk value which is086461113957σk In the following we shall express RMSEk in theunit of 1113957σk for simplicity In Figure 1 we illustrate the

RMSEk values as functions of κ1 and κ2 for the case ofϕ 35deg In this figure we note that the RMSEk value rises as(κ1 κ2) moves away from (07 05) In particular we notethat the RMSEk value rises relatively mildly if κ1 or κ2 isdecreased However the RMSEk value rises significantly asκ1 and κ2 approach 1 We found that the RMSEk valuesgenerally rise slowly as the ϕ value moves away from 35degAlthough it is not possible to include all results with dif-ferent ϕ values we include the results with ϕ 15deg and ϕ

55deg in Figures 2 and 3 to illustrate how the RMSEk valuechanges with the ϕ value

So far we discussed how we can choose the optimalobserver gain vector with (45) assuming that σk[n] is aconstant independent of n Hence the validity of the pro-posed method of obtaining the observer gain will depend onhow the effect of variations in σk[n] affects the value ofRMSEk According to our extensive test the effect of vari-ation of σk[n] is very negligible except for extreme situationsthat do not affect the optimal selection of the observer gainTo illustrate in a space economical way we consider a sit-uation in which there are five objects with DoA trajectoriesillustrated in Figure 4 In realistic environments the DoAtrajectories may not change as fast as in Figure 4 However ifwe choose a slowly moving environment the σk[n] valuewould not change very much and hence the effect wouldalso be negligible For this reason we have chosen a very fastchanging environment shown in Figure 4 To obtain σk[n]we assume that only object k moves following the trajectoryθk(nT) and perform simulations with the proposed scheme(DSPCLT) We recall that we divided the situations intoresolvable and nonresolvable situations to avoid overlappingsituations For this reason we assume that only object k

exists in computing σk[n]For numerical evaluations we assume that the number

N of sensor elements is 12 the total number NT of ob-servation intervals is 60 and the signal-to-noise ratio is 5 dBWith 106 simulation run trials we computed σ2k[n] which isillustrated in Figure 5 In this figure we note that the value ofσ2k[n] is generally related to the angular acceleration Afterobtaining the estimate of σ2k[n] we computed RMSEk using

(44) in the unit of 1113957σk

(1NT) 1113936NT

n1 σ2k[n]

1113969

To compare theresults with the values computed using (47) we evaluatedthe ratio obtained by dividing the RMSEk value computedusing (44) by that using (47) for each set of κ1 κ2 and ϕvalues Some results of the evaluations are summarized inTable 1 In this table the number pair (a b) means the


minimum and maximum values of the ratios over all κ1 andκ2 values for given ϕ value and sequence σ2k[n] For examplefor ϕ 15deg and σ21[n] the ratio falls between 0965 and 1141With this set of numbers we can conclude how different thetwo RMSEk values computed by (42) and (45) are to theextreme However the extreme values do not tell the wholestory since the extreme values namely the maximum andminimum ratio values occur at κ1 and κ2 values close to 1 sothat the RMSEk value is very large In other words theextreme values happen with eigenvalue selections for whichek[n] do not converge to zero very fast

To understand what happens let us first consider thecase of ϕ 15deg and σ25[n] in which the minimum value isparticularly small In Figure 6 we plotted the ratios as afunction of κ1 and κ2 for this case namely for ϕ 15deg andσ25[n] We observe that the minimum value 0799 happensnear the region κ1 asymp 1 and κ2 asymp 1 for which the RMSEk

values are very large In contrast the ratio is very close to 10except for the region in which RMSEk remains to be ac-ceptably small Moreover we recall that the results in Fig-ure 6 correspond to some extreme case in which the ratiofluctuates a lot As a more typical case we depict in Figure 7

02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

088

091

087 088 091

152

088087

087088 09

Figure 1 RMSEk for ϕ 35deg as a function of κ1 and κ2

02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

091

152

5

1

215

09

1


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

09

088

1

215

1

09

1

Figure 3 RMSEk for ϕ 55deg as a function of κ1 and κ2D

oA (deg

)

20 30 40 50 6010Time index n

0

30

60

90

120

150

180

θ1 (nT)θ2 (nT)θ3 (nT)

θ4 (nT)θ5 (nT)

Figure 4 DoA trajectories used for simulations


the ratio for ϕ 35deg and σ21[n] We note that the ratio isagain very close to 10 except for the region in which κ1 asymp 1and κ2 asymp 1 Consequently we observe that the RMSEk value

computed with (47) is very close to that with (44) except atκ1 κ2 and ϕ values for which ek[n] does not converge to zerofast enough and RMSEk is very large

20 30 40 50 60100Time index n

σ12[n]

σ22[n]

σ32[n]

σ42[n]

σ52[n]

σ2 k[n]

0

05

1

15

2

25

Figure 5 σ2k[n] in the unit of 1113957σk

Table 1 (Minimum maximum) pair of the RMSEk value ratio

σ21[n] σ22[n] σ23[n] σ24[n] σ25[n]

ϕ 15deg (0965 1141) (0973 1022) (0842 0984) (0908 0980) (0799 0987)

ϕ 35deg (0965 1124) (0973 1017) (0859 0984) (0916 0980) (0822 0987)

ϕ 55deg (0967 1050) (0973 0995) (0922 0984) (0949 0980) (0908 0987)

02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09 098

095 085

08

098

09509

085

098

09509

Figure 6 RMSE5 computed with time-varying σ5[n] for ϕ 15deg

02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

097

097

0971

1

1

105

105

11



7 System Performance andComputational Complexity

In the section we compare the performance and thecomputational complexity of the proposed scheme withthose of two existing algorithms namely AMEND andPASTd Kalman Most of all we observe that the proposedscheme (DSPCLT) exhibits excellent performance withsignificantly low computational complexity in comparisonwith AMEND and PASTd Kalman

71 Performance Evaluation of Algorithms First let uscompare the performance of DSPCLT with that of AMENDand PASTd Kalman Asmeasures of performance we use thefollowing quantities

RMSEk[n]

E 1113954θk(nT) minus θk(nT)11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

1113970

(49)

RMSET

1KNT

1113944

K

k11113944

NT

n1E 1113954θk(nT) minus θk(nT)

11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(50)

where 1113954θk(nT) is the estimated DoA of θk(nT) and NT is thethe total number of observation intervals We note thatRMSEk[n] is the RMSE of the DoA of the kth object at timet nT and that RMSET represents the total RMSE over allobjects and observation intervals We recall that we regard1113954θk[n | n] as 1113954θk(nT) in the proposed scheme

However we need to be very careful in assessing theperformance with RMSE is is because there are somepossibilities of tracking wrong objects after passing througha crossover point or of object tracking divergence due to lackof sensitivity to object movements which can result in hugeestimation errors e value of RMSE can be meaninglesswhen such catastrophic tracking failure happens To makethe value of RMSE more meaningful we shall regard thesituation as tracking failure if and only if the estimated DoAof any of the signals is more than 5deg away from the actualDoA and compute the RMSE value disregarding such casesof tracking failure (In this paper we shall measure DoAsand RMSE in degrees rather than in radians) Since theRMSE values do not account for the cases of tracking failurewe shall also consider in addition to RMSE the probabilityPF of tracking failure as a performance measure

For performance evaluations we first consider a situa-tion in which K 3 signals impinge upon a ULA with N

12 sensors separated by half-wavelength from time-varyingdirections θ1(nT) θ2(nT) and θ3(nT) n 1 2 NT 60 illustrated in Figure 8 We assume that the signal-to-noise ratio of each signal at each sensor is given as 5 dBDuring each time interval [nT (n + 1)T) total Ns 200data snapshots are uniformly captured at each sensor withδ 1Ns

We note that the choice of forgetting factor ρ affects thetracking performance Hence we compare the performancewith various ρ values from 091 to 099 In the case of PASTdKalman we also consider instead of ρNsminus n the forgetting

coefficient n(n + 1) which was used in [21] For AMENDwe use the observer gain vector obtained with the eigen-values 075 075ejπ18 and 075eminus jπ18 as suggested in [22]However we select the observer gain vector obtained usingLemma 2 with eigenvalues 07 05ej35π180 and 05eminus j35π180

for DSPCLT We recall that observer gain is not used inPASTd Kalman

As the value of M is used to compute 1113957rm in (A3) and(A4) we use M K since the RMSE performance enhancesrapidly as M increases from 1 to K and then becomesvirtually invariant as it further grows Also we separate theresolvable and nonresolvable situations with thresholdvalues 5deg 45deg and 4deg for N 12 16 and 20 respectively

Simulations are performed with 105 trials e failureprobability PF and RMSET obtained through the simulationsare summarized in Table 2 We may say that AMEND ex-hibits the most reliable performance in that it does notexperience failure for any choice of ρ value HoweverAMEND is worst in terms of RMSET value In particular wenote that both DSPCLTand PASTd Kalman exhibit not onlybetter RMSET values than AMEND but also achieve zerofailures e best RMSET value for each scheme is marked inbold-faced fonts and we note that DSPCLT achieves thelowest RMSET value among the three schemes By the waythe RMSET value 06160 listed on the lowest row for PASTdKalman corresponds to the forgetting coefficient n(n + 1)

suggested in [21]To illustrate the performance variation over time

RMSEk[n] is plotted in Figure 9 In this figure we used ρ

095 for AMEND and 097 for DSPCLTand PASTd KalmanIn Table 2 we note that AMEND exhibits the worst RMSET

values Also we note DSPCLT achieves slightly betterRMSET value compared with PASTd Kalman In Figure 9we observe that DSPCLT exhibits best overall RMSEk[n]

performance except for the region where the DoAs of objects

20 30 40 50 6010Time index n


DoA

(deg)

0

30

60

90

120

150

180



overlap In particular we note that the performance ofDSPCLT is significantly better than others for the first objectduring n isin 1 2 middot middot middot 10 From Figure 8 we note that theDoA of the first object changes very fast overn isin 1 2 middot middot middot 10 Consequently we can conclude that thedelay compensation algorithm introduced in Section 52indeed provides improved performance in fast changingenvironments We also observe that DSPCLT is less sus-ceptible to object crossover compared with AMEND is isbecause the nonresolvable situation is separately treated inthe proposed scheme using the algorithm described inSection 4

One of the main reasons for the overall performancesuperiority of the proposed scheme is the comparativelylarger effective aperture size with the same number of sensorelements For illustration of the results of effective aperturesize we compare in Figure 10 realizations of the functionf(θ) defined in (11) at time index n 19 In this figure weobserve that f(θ) is much sharper for DSPCLT than forAMEND and PASTd Kalman For this reason DSPCLTprovides better overall RMSE performance than AMENDrivaling PASTd Kalman that employs much more sophis-ticated Kalman filter algorithm

We next study the performance variation with increasednumber of objects For illustration we consider the situationin which there are K 5 objects with trajectories illustratedin Figure 4 Here we note that AMEND can track only K

objects satisfying KltN2 while DSPCLT and PASTdKalman can track upto N minus 1 objects Consequently at leastN 11 sensor elements are needed for AMEND to be able totrack K 5 objects To see the effect of the number of sensorelements we consider three values 12 16 and 20 as thenumber N of antenna elements of the ULA Again thereceived SNR of each of the 5 objects is set to be 5 dB thenumber Ns of snapshots is chosen to be 200 with δ 1Nsand the value of M is chosen to be the same as K

e simulations are performed with 105 runs again andthe results are summarized in Table 3 and Figure 11 As inthe case of K 3 we obtained simulation results withvarious forgetting factor values and included the best resultsin the table and the figure In particular the value ρ indicatesthe forgetting factor value that leads to the best RMSET

value In Table 3 and Figure 11 we observe that AMENDexhibits the worst overall performance in almost every as-pect We observe that both AMEND and DSPCLT experi-enced nonzero failure probabilities which decrease as thenumber of sensor elements increases Consequently we canconclude that PASTd Kalman achieves the most reliableperformance among the three considered schemes How-ever we note that DSPCLTexhibits performance better thanPASTd Kalman with a sufficiently large number of sensorelements We also note that DSPCLT compared withAMEND not only achieves lower failure probability but alsoexhibits lower RMSET In particular we observe that theRMSET value of DSPCLT is significantly lower than that ofAMEND despite the fact that RMSET is computed excludingthe cases in which tracking fails Next in Figure 11 thesequence of RMSEk[n] values for each of the five objects isdepicted for the case of N 20 As in the previous case ofK 3 objects AMEND exhibits the worst performance andDSPCLT achieves similar or better performance comparedwith PASTd Kalman except around crossover points

72 Computational Complexity In this section we comparethe computational complexities of DSPCLT AMEND andPASTd Kalman Before proceeding we note that the com-plexity of DSPCLTvaries depending on whether the object isin the resolvable or nonresolvable situation Since thecomplexity of DSPCLT in the nonresolvable situation is

Table 2 PF and RMSET (K 3 N 12 M 3 Ns 200)

ρDSPCLT AMEND PASTd Kalman

PF[]RMSET

[deg]PF[]

RMSET

[deg]PF[]

RMSET

[deg]

091 040 04354 0 04719 083 03450092 024 04155 0 04634 057 03276093 012 03742 0 04561 030 03084094 009 03541 0 04510 010 02928095 005 03181 0 04503 003 02792096 003 03014 0 04602 002 02831097 0 02715 0 04905 0 03247098 0 02923 0 05713 0 04308099 0 03851 0 07496 0 06328mdash mdash mdash mdash mdash 0 06160

RMSE3[n]

20 30 40 5010Time index n

AMENDPASTd KalmanDSPCLT

06

12

RMSE

(deg )

RMSE2[n]


06

12

RMSE

(deg )

RMSE1[n]


06

12

RMSE

(deg )

Figure 9 Performance variation over time N 12 K 3


significantly smaller than that in the nonresolvable situationand nonresolvable situation happens only very rarely whenthe number of sensor elements is large enough we shallassume that only resolvable situation happens when com-puting the complexity of DSPCLT Before proceeding wenote that all the three tracking schemes DSPCLT AMENDand PASTd Kalman can be divided into three processesnamely ldquosubspace constructionrdquo ldquoprojector constructionrdquoand ldquofilteringrdquo Here the subspace construction is in essencethe construction of the vectors that generate the signalsubspace

In PASTd Kalman [21] and AMEND [22] subspaceconstruction is carried out using the recursive least square(RLS) method while the vectors in DSPCLTare constructeddirectly from the estimated correlationmatrix using (A1) Inthe projector construction stage an orthogonal projectormatrix such as Π[n] is constructed And state vectors areupdated using Kalman filter or Luenberger observer in thefinal stage of filtering As described in Section 1 Kalmanfilter is used in PASTd Kalman and Luenberger observer isused in DSPCLT and AMEND In Table 4 the numbers ofreal floating point operations needed to carry out each of thethree processes for each of the three schemes as well as thetotal numbers are presented

For more concrete picture of computational complexitywe evaluated the numbers in Table 4 with the choice ofparameters used in obtaining Figures 9 and 11 the results ofwhich are presented respectively in Tables 5 and 6 We notethat a significant portion of computational complexity isneeded in subspace construction We note that AMENDrequires particularly large numbers of operations in thesubspace construction compared with DSPCLT is isbecause AMEND uses matrix inversion and QR decom-position even though it does not use ED nor SVD We notethat DSPCM does not require matrix inversion nor QRdecomposition We note that PASTd Kalman comparedwith DSPCLT and AMEND requires larger amount ofoperations in the filtering stage since it uses computationallyexpensive Kalman filtering Finally we note that DSPCLTrequires 2 sim 6 times more operations to carry out projectorconstruction and filtering compared with AMEND is isbecause DSPCLT has larger effective aperture size despitethat the same number of sensors is used However theoverall computational complexity of DSPCLT is significantlysmaller than that of AMEND and PASTd Kalman becausethe computational complexity of the second stage is muchsmaller than that of either the first or the third stage OverallDSPCLT requires only about 1199 (1878) and 770

f (θ)

40 60 80 100 120 140 16020θ (deg)

0

5

10

15

20

DSPCLTPASTd KalmanAMEND

Figure 10 Comparison of f(θ) functions at time index n 19

Table 3 PF and RMSET (K 5 M 5 Ns 200 N 12 16 20)

NDSPCLT AMEND PASTd Kalman

ρ PF[] RMSET [deg] ρ PF[] RMSET [deg] ρ PF[] RMSET [deg]

12 097 441 05080 097 2216 11589 097 0 0378716 097 004 02791 095 031 04360 097 0 0313520 095 0 01884 093 0 02937 096 0 02433


Table 4 Required floating point operations

DSPCLT AMEND PASTd KalmanSubspace construction 5M(2N minus K + 1)Ns 64K2(N minus 2K)Ns K(28N + 7)Ns

Projector construction 8K(2N minus K)2 16(N minus K)(N minus 2K)2 8KN2

Filtering 16(2N minus K)2 16(N minus K)2 8N2Ns + 8(K + 2)N2

Total 5M(2N minus K + 1)Ns

+(8K + 16)(2N minus K)264K2(N minus 2K)Ns+

16(N minus K)[(N minus 2K)2 + N minus K](28KN + 8N2 + 7K)Ns + 16(1 + K)N2

Table 5 Required floating point operations N 12 M K 3 Ns 200

DSPCLT AMEND PASTd KalmanSubspace construction 66 000 691 200 205 800Projector construction 10 584 5 184 3 456Filtering 7 056 1 296 236 160Total 83 640 697 680 445 416


DSPCLT AMEND PASTd KalmanSubspace construction 180 000 3 200 000 567 000Projector construction 49 000 24 000 16 000Filtering 19 600 3 600 662 400Total 248 600 3 227 600 1 245 400



0408

RMSE

(deg) RMSE5[n]


0408

RMSE

(deg) RMSE4[n]


0408

RMSE

(deg) RMSE3[n]


0408

RMSE

(deg) RMSE2[n]


0408

RMSE

(deg) RMSE1[n]



(1996) real floating point operations compared withAMEND and PASTd Kalman respectively for the case ofN 12 M K 3 Ns 200 and N 20 M K 5

Ns 200

8 Conclusion

In this paper we proposed the direct signal space con-struction method Luenberger tracker with quadratic leastsquare regression for computationally efficient DoA track-ing Also we studied analytically how the selection of ob-server gain value affects the system performance anddescribed how we can use the theoretical results to optimizethe observer gain value We note that the proposed algo-rithm combines the direct signal space construction methodand Luenberger observer to achieve computational com-plexity significantly lower than other schemes Also thealgorithm achieves enhanced performance by employing amethod of delay compensation which accounts for thedifference in the times of estimation and data collection eproposed scheme avoids the possible issue of non-resolvability by treating separately the situation of objectoverlap As a result the proposed scheme achieves perfor-mance similar or superior to existing algorithms with sig-nificantly lower computational complexity which wasverified with a variety of simulation results

Appendix

A Orthogonal ProjectorConstruction with DSPCM

In this paper we propose to use a method of constructing anorthogonal projector based on DSPCM in [14 15] We firstdescribe how the subspace and the orthogonal projector aredefined en we describe how we obtain an estimate of theorthogonal projector is computed based on the data col-lected at antenna sensor elements

A1 Orthogonal Projector Based on DSPCM Before discus-sing the concept of the orthogonal projector we need to definethe subspace we are going to work with For this purpose wefirst define the signal-only correlation matrix Rs by

Rs(t) equiv E x(t)x(t)H

1113960 1113961 minus N0IN (A1)

where the value of noise power spectral density N0 shall beassumed to be estimated with long-term observation ofsignal-free data collected at the sensors It is not difficult toshow by inserting (1) into (49) that Rs(t) APAH andhence that Rs(t) is a Hermitian Toeplitz matrix with its(m n) element given by Rmn(t) 1113936

Kk1 pkej(mminus n)]k(t) Due to

the Toeplitz structure there are at most 2N minus 1 distinctelements inRs(t) namelyRmN Rmminus 1N middot middot middot RNN RNNminus 1 and RN1 For simplicity of notation we shall denote these2N minus 1 elements of Rs by r1(t) r2(t) middot middot middot r2Nminus 1(t) We notethat rm(t) satisfies

rm(t) 1113944K

k1pke

j(minus N+m)]k(t) m 1 2 middot middot middot 2N minus 1

(A2)

and contains information about the DoAs namelyθ1(t) middot middot middot θK(t)

While traditional subspace-based algorithms such asMUSIC or ESPRITattempt to define signal or noise subspaceby making orthogonal decomposition of the given vectorspace DSPCM attempts to define not by a subspace of agiven vector space but by rearranging the elements of Rs Tobe more specific we define K signal vectors r1(t) middot middot middot rK(t)

by

rk(t) rk(t) rk+1(t) middot middot middot rk+2Nminus Kminus 1(t)1113858 1113859T k 1 2 middot middot middot K

(A3)

and regard as the signal space at time t the vector spacespanned by these K signal vectors If we letV as the complexvector space consisting of all L 2N minus K-dimensionalcolumn vectors then the signal space is a subspace ofV Weshall call the orthogonal complement of the signal spacewithin V the noise space In the first and the third stage ofthe Luenberger observer we need the orthogonal projectoronto the noise subspace which we shall denote by Π(t)From the knowledge of elementary linear algebra [28] it isnot difficult to see that Π(t) can be written as

Π(t) IL minus 1113954r1(t)1113954rH1 (t) + middot middot middot + 1113954rK(t)1113954rH

K (t)1113872 1113873 (A4)

where the vectors 1113954r1(t) middot middot middot 1113954rK(t) are the orthonormalvectors obtainable by applying GramndashSchmidt process onr1(t) middot middot middot rK(t) From the results in [14 15] we note that θequals to one of θ1(t) middot middot middot θK(t) if and only ifaH(θ)Π(t) a(θ) 0 which can be used to make initialestimates of the DoAs

A2 Estimation of the Orthogonal Projector Next to buildan estimate 1113954Π[n] of the projector Π(nT) data samples attimes between (n minus 1)T and nT are used In particular weassume that there are total Ns available samples of xk(t)for each k 1 2 middot middot middot N snapshotted at times t

[n minus (Ns minus 1)δ]T [n minus (Ns minus 2)δ]T middot middot middot [n minus (Ns minus Ns)]THere δ is a certain positive real number such that Nsδ le 1For notational simplicity we shall write xkl forxk(nT minus (Ns minus l)δT) From these data samples the esti-mates 1113957rm of rm in (50) are computed by

1113957rm 1m

1113944

m

k11113944

Ns

l1

(1 minus ρ)ρNsminus l

1 minus ρNs

xklxlowastNminus m+kl (A5)

for m 1 2 middot middot middot M and

1113957rm 1

M1113944

M

k11113944

Ns

l1


1 minus ρNs

xklxlowastNminus m+kl minus N0δNm (A6)

for m M M + 1 middot middot middot N Here M is a positive integer notexceeding N and ρ denotes a forgetting factor chosen tosatisfy 0lt ρlt 1 We then obtain 1113957rN+1 middot middot middot 1113957r2Nminus 1 from


1113957r1 middot middot middot 1113957rNminus 1 using the relation1113957r2Nminus m 1113957rlowastm m 1 2 middot middot middot N minus 1 Next we obtain K ortho-normal vectors 1113954r1 middot middot middot 1113954rK by applying GramndashSchmidt pro-cedure on 1113957r1 middot middot middot 1113957rK where 1113957rkrsquos are N-dimensional columnvectors defined by 1113957rk [1113957rk 1113957rk+1 middot middot middot 1113957rk+Lminus 1]

T Next weobtain 1113954Π[n] by

1113954Π[n] IL minus 1113954r11113954rT1 + middot middot middot + 1113954rK1113954rT

K1113872 1113873 (A7)

B Proof of Theorem 1

First using assumptions (42) and (43) we obtain

E cTek[n]11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877 E cTVΛnVminus 1ek[0]+ 1113944

n

m1cTVΛnminus m


111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

cTVΛnVminus 1E ek[0]eT

k [0]1113960 1113961 VΛnVminus 11113872 1113873

Tc

+ 1113944n

m1cTVΛnminus m

(I minus Λ)Vminus 1c1113872 11138732E εk[m]εk[m]1113858 1113859

(B1)

Next to compute E[ek[0]eTk [0]] we note that

ek[0] 1113954yk[0 | 0] minus yk(0)

1113957θk[0 | 0] minus θk(0)

1113957θk[0 | 0] minus 1113957θk[minus 1 | minus 1]

Tminusθk(0) minus θk(minus T)

T

1113957θk[0 | 0] minus 21113957θk[minus 1 | minus 1] + 1113957θk[minus 2 | minus 2]

T2 minusθk(0) minus 2θk(minus T) + θk(minus 2T)

T2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B2)

εk[0]

εk[0] minus εk[minus 1]Tεk[0] minus 2εk[minus 1] + εk[minus 2]T2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B3)

Now by directly plugging (B3) into E[ek[0]eTk [0]] and

using (43) we obtain

E ek[0]eTk [0]1113960 1113961

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B4)

which completes the proof

Data Availability

e data used to support the findings of this study are in-cluded in the two supplementary information files

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is research was supported by Basic Science ResearchProgram through the National Research Foundation ofKorea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B07045719) and Korea Electric Power Cor-poration (Grant number R18XA02)

Supplementary Materials

e data used to support the findings of this study are in-cluded within the supplementary information files namedlsquoDoA_kxlsxrsquo and lsquosigma2_kxlsxrsquo (1) lsquoDoA_kxlsxrsquo includes


the numerical values of θk(nT) in Figures 4 and 8 (2)lsquosigma2_kxlsxrsquo includes the numerical values of σ2k[n] inFigure 5 (Supplementary Materials)

References

[1] R Schmidt ldquoMultiple emitter location and signal parameterestimationrdquo IEEE Transactions on Antennas and Propagationvol 34 no 3 pp 276ndash280 1986

[2] R Roy and T Kailath ldquoEsprit-estimation of signal parametersvia rotational invariance techniquesrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 37 no 7pp 984ndash995 1989

[3] A Barabell ldquoImproving the resolution performance ofeigenstructure-based direction-finding algorithmsrdquo in Pro-ceedings of the ICASSPrsquo83 IEEE International Conference onAcoustics Speech and Signal Processing vol 8 pp 336ndash339Boston MA USA April 1983

[4] P Pal and P P Vaidyanathan ldquoNested arrays a novel ap-proach to array processing with enhanced degrees of free-domrdquo IEEE Transactions on Signal Processing vol 58 no 8pp 4167ndash4181 2010

[5] P Pal and P P Vaidyanathan ldquoCoprime sampling and themusic algorithmrdquo in Proceedings of the 2011 Digital SignalProcessing and signal Processing Education meeting (DSPSPE) IEEE Sedona AZ USA pp 289ndash294 January 2011

[6] C El Kassis J Picheral G Fleury and C Mokbel ldquoDirectionof arrival estimation using em-esprit with nonuniform ar-raysrdquo Circuits Systems and Signal Processing vol 31 no 5pp 1787ndash1807 2012

[7] K Huang J Sha W Shi and Z Wang ldquoAn efficient fpgaimplementation for 2-d music algorithmrdquo Circuits Systemsand Signal Processing vol 35 no 5 pp 1795ndash1805 2016

[8] J Chen S Guan Y Tong and L Yan ldquoTwo-dimensionaldirection of arrival estimation for improved archimedeanspiral array with music algorithmrdquo IEEE Access vol 6pp 49740ndash49745 2018

[9] X Wang W Wang and D Xu ldquoLow-complexity ESPRIT-root-MUSIC algorithm for non-circular source in bistaticMIMO radarrdquoCircuits Systems and Signal Processing vol 34no 4 pp 1265ndash1278 2015

[10] J Munier and G Y Delisle ldquoSpatial analysis using newproperties of the cross-spectral matrixrdquo IEEE Transactions onSignal Processing vol 39 no 3 pp 746ndash749 1991

[11] S Marcos A Marsal and M Benidir ldquoe propagatormethod for source bearing estimationrdquo Signal Processingvol 42 no 2 pp 121ndash138 1995

[12] A Eriksson P Stoica and T Soderstrom ldquoOn-line subspacealgorithms for tracking moving sourcesrdquo IEEE Transactionson Signal Processing vol 42 no 9 pp 2319ndash2330 1994

[13] N Xi and L Liping ldquoA computationally efficient subspacealgorithm for 2-d doa estimation with l-shaped arrayrdquo IEEESignal Processing Letters vol 21 no 8 pp 971ndash974 2014

[14] D-S Yoo ldquoSubspace-based doa estimation with sliding sig-nal-vector construction for ulardquo Electronics Letters vol 51no 17 pp 1361ndash1363 2015

[15] D-S Yoo ldquoA low complexity subspace-based doa estimationalgorithm with uniform linear array correlation matrixsubsamplingrdquo International Journal of Antennas and Prop-agation vol 2015 Article ID 323545 10 pages 2015

[16] B Yang ldquoProjection approximation subspace trackingrdquo IEEETransactions on Signal Processing vol 43 no 1 pp 95ndash1071995

[17] G Xu H Zha G Golub and T Kailath ldquoFast algorithms forupdating signal subspacesrdquo IEEE Transactions on Circuits andSystems II Analog and Digital Signal Processing vol 41 no 8pp 537ndash549 1994

[18] C K Sword M Simaan and E W Kamen ldquoMultiple targetangle tracking using sensor array outputsrdquo IEEE Transactionson Aerospace and Electronic Systems vol 26 no 2 pp 367ndash373 1990

[19] K W Lo and C K Li ldquoAn improved multiple target angletracking algorithmrdquo IEEE Transactions on Aerospace andElectronic Systems vol 28 no 3 pp 797ndash805 1992

[20] J M Goldberg ldquoJoint direction-of-arrival and array-shapetracking for multiple moving targetsrdquo IEEE Journal of OceanicEngineering vol 23 no 2 pp 118ndash126 1998

[21] J Sanchez-Araujo and S Marcos ldquoAn efficient pastd-algo-rithm implementation for multiple direction of arrivaltrackingrdquo IEEE Transactions on Signal Processing vol 47no 8 pp 2321ndash2324 1999

[22] J Xin N Zheng and A Sano ldquoSubspace-based adaptivemethod for estimating direction-of-arrival with luenbergerobserverrdquo IEEE Transactions on Signal Processing vol 59no 1 pp 145ndash159 2010

[23] R E Kalman ldquoA new approach to linear filtering and pre-diction problemsrdquo Journal of Basic Engineering vol 82 no 1pp 35ndash45 1960

[24] D Luenberger ldquoAn introduction to observersrdquo IEEETransactions on Automatic Control vol 16 no 6 pp 596ndash6021971

[25] J Xin and A Sano ldquoComputationally efficient subspace-basedmethod for direction-of-arrival estimation without eigende-compositionrdquo IEEE Transactions on Signal Processing vol 52no 4 pp 876ndash893 2004

[26] M Wax and T Kailath ldquoDetection of signals by informationtheoretic criteriardquo IEEE Transactions on Acoustics Speechand Signal Processing vol 33 no 2 pp 387ndash392 1985

[27] J Makhoul ldquoLinear prediction a tutorial reviewrdquo Proceedingsof the IEEE vol 63 no 4 pp 561ndash580 1975

[28] S Lang Linear Algebra Springer Berlin Germany 3rdedition 2004


recently the authors in [22] proposed an algorithm calledthe adaptive method of estimating DoA (AMEND) whichreplaces the Kalman filter with Luenberger state observer[24] However the complexity of AMEND is not smallerthan PASTd Kalman because it employs for subspaceconstruction a relatively computationally expensive algo-rithm called the subspace-based method without eigende-composition (SUMWE) [25]

In this paper we propose DSPCLT that achieves similaror better performance with computational complexitysignificantly lower than existing algorithms e proposedscheme has several distinct features compared with existingschemes First it achieves very low complexity byemploying DSPCM [14 15] for subspace construction andLuenberger observer in place of the Kalman filter for es-timation and filtering Secondly it achieves robustness bytreating separately the special case of object overlap inwhich subspace-based algorithms suffer from the lack ofresolvability irdly the proposed scheme achieves en-hanced performance by a method of delay compensationwhich takes into account the fact that the DoA estimationat present time is performed with observed data collectedpreviously We show through numerical analysis that theproposed scheme achieves performance similar or superiorto existing algorithms with only a fraction of computationalrequirement

Another important topic studied in this paper is thechoice of observer gains used to implement Luenbergerobserver In most existing schemes employing Luenbergerobserver optimal observer gain values are determinedthrough simulations For example a certain observer gainvalue was suggested in [22] after simulations However wenote that there is infinite variety of situations depending onthe number of antenna sensor elements the numbers ofobjects the speeds of movements the signal-to-noise ratiosof the target the shape of object trajectory and so on Hencefor a selection of a certain observer gain value to be acceptedas a valid choice there must be some degree of guaranteethat a particular observer gain value will provide optimal ornear-optimal performance in other situations which isgenerally very difficult with pure simulations For thisreason we study analytically how the choice of observer gainaffects the algorithm performance and how we can chooseoptimal observer gain value

is paper is organized as follows In Section 2 thesystem model and basic assumptions are described to-gether with the notational definitions used throughoutthis paper In Sections 3 sim 5 the proposed scheme(DSPCLT) is delineated which consists of three stagescalled initialization prediction and estimatione stagesof initialization and prediction are introduced in Section 3and that of estimation is presented in Sections 4 and 5depending on whether objects are resolvable by thesubspace-based algorithm employed In Section 6 westudy theoretically how the selection of observer gainvalue affects the algorithm performance and describe howwe can make optimal observer gain value selection usingthe analytical results Next we show in Section 7 that theproposed scheme achieves similar or better performance

with only a fraction of computational requirementscompared with other algorithms Finally we draw con-clusion in Section 8

e following notations are used throughout this paperBold-faced letters are used for matrices and vectors ecomplex conjugation the transposition and the Hermitiantransposition of matrix A are denoted respectively by AlowastAT and AH e M times N zero matrix and N times N identitymatrix are denoted respectively by OMtimesN and IN

2 System Model and Basic Assumptions

We consider a uniform linear array (ULA) consisting of N

sensors with intersensor spacing d and assume that K

narrowband far-field signal waves s1(t) s2(t) middot middot middot sK(t) ofwavelength λ impinge on the array at distinct directionsrespectively making angles θ1(t) θ2(t) middot middot middot θK(t) with thelinear array We denote by x1(t) x2(t) middot middot middot xN(t) the re-ceived noisy signals collected at the N sensors en if wedefine the steering vector a(θk(t)) by a(θk(t))

[1 ej]k(t) middot middot middot ej(Nminus 1)]k(t)]T with ]k(t) 2πd cos θk(t)λ thesystem can be modeled with the following equation

x(t) A θ1(t) middot middot middot θK(t)( 1113857s(t) + n(t) (1)

where x(t) [x1(t) x2(t) middot middot middot xN(t)]T A(θ1(t) middot middot middot

θK(t)) [a(θ1(t)) middot middot middot a(θK(t))] s(t) [s1(t) middot middot middot

sK(t)]T and n(t) is the vector consisting of the noises addedat the sensors In this paper we make on the system modelthe following basic assumptions

(1) e signal wavelength λ is known a priori and that d

is chosen to be λ2 And the number K of impingingsignals is predetermined by appropriate methodssuch as the one in [26]

(2) e vectors s(t) and n(t) are statistically indepen-dent complex random vector processes with zeromean satisfying

E s(t)s(s)H

1113960 1113961 Pδts

E s(t)s(s)T

1113960 1113961 OKtimesK

E n(t)n(s)H

1113960 1113961 N0δtsIN

E n(t)n(s)T

1113960 1113961 ONtimesN

(2)

where δts is the Kronecker delta function We furtherassume that P is a diagonal matrix with positive realnumbers p1 middot middot middot pK on its diagonal and that thevalue of N0 is assumed to be estimated accurately bythe receiver through long-term observation of thenoise

(3) e system tracks the DoAs by obtaining angle es-timates at time t 0 T 2T 3T where T is apredefined positive real number We assume furtherthat T is chosen to be small enough that the DoAs areessentially constant during the time duration [(n minus

1)T nT] for each n 1 2 3















where

F

1 T 05T2

0 1 T

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (5)



11138681113868111386811138681113868 (6)










a

b

c

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p 1113936minus 1

mminus p

m 1113936minus 1

mminus p

m2

1113936minus 1

mminus p

m 1113936minus 1

mminus p

m2 1113936minus 1

mminus p

m3

1113936minus 1

mminus p

m2 1113936minus 1

mminus p

m3 1113936minus 1

mminus p

m4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 1

middot

1113936nminus 1

mnminus p

1113954θk[m | m]

1113936nminus 1

mnminus p


1113936nminus 1

mnminus p

(m minus n)21113954θk[m | m]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(8)


1113954yk[n | n]

c

b

2a

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (9)














fPrime(θ) θ minus

Re aprimeH(θ)Πa(θ)1113876 1113877


(12)




dH[n] 1113954Π[n]d[n] (13)



ak[n] a 1113954θk[n | n minus 1]1113872 1113873 (14)

and

dk[n] da(θ)

dθ

111386811138681113868111386811138681113868 θ1113954θk[n | nminus 1] (15)










1 minus ρNs


1 minus ρNs


1 minus ρNs



(16)




11 minus α





(19)




(20)



Ns minus Ns + 1( 1113857ρ + ρNs+11113858 1113859


(21)















(25)



(26)





(27)










RMSEk

1NT

1113944

NT


11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(30)


RMSEk

1NT

1113944

NT

n1E cTek[n]

11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(31)



cTek[n](32)


Lemma 1










gk1 1 minus κ1κ22

gk2 12T


gk3 1


(36)




λ3 + gk1 + gk2T + 05gk3T2

minus 31113872 1113873λ2


+ 31113872 1113873λ + gk1 minus 1 0(37)


λ3 + gk1 + gk2T + 05gk3T2

minus 31113872 1113873λ2


+ 31113872 1113873λ + gk1 minus 1


(38)





(39)



m1VΛnminus m


+ 1113944n

m1VΛnminus m


1113960 1113961δk[m]

(40)





m1VΛnminus m


(41)



E εk[n]1113858 1113859 0 (42)

and that

E εk[m]εk[n]1113858 1113859 δmnσ2k[n] (43)

en

RMSEk

1NT

1113944

NT


1113872 1113873Tc + 1113944

n

m1cTVΛnminus m


11139741113972

(44)

where

T

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(45)




σk[n] 1113957σk (46)

for all n then

RMSEk 1113957σk

1NT

1113944

NT


1113872 1113873Tc + 1113944

n

m1cTVΛnminus m

(I minus Λ)Vminus 1c1113872 11138732

⎡⎣ ⎤⎦

11139741113972

(47)

where

T

11T

1T2

1T

2T2

3T3

1T2

3T3

6T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(48)


(1NT) 1113936NT

n σ2k[n]

1113969










(1NT) 1113936NT

n1 σ2k[n]

1113969






02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

088

091

087 088 091

152

088087

087088 09


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

091

152

5

1

215

09

1


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

09

088

1

215

1

09

1


oA (deg

)

20 30 40 50 6010Time index n

0

30

60

90

120

150

180


θ4 (nT)θ5 (nT)





20 30 40 50 60100Time index n

σ12[n]

σ22[n]

σ32[n]

σ42[n]

σ52[n]

σ2 k[n]

0

05

1

15

2

25



σ21[n] σ22[n] σ23[n] σ24[n] σ25[n]

ϕ 15deg (0965 1141) (0973 1022) (0842 0984) (0908 0980) (0799 0987)

ϕ 35deg (0965 1124) (0973 1017) (0859 0984) (0916 0980) (0822 0987)

ϕ 55deg (0967 1050) (0973 0995) (0922 0984) (0949 0980) (0908 0987)

02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09 098

095 085

08

098

09509

085

098

09509


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

097

097

0971

1

1

105

105

11






RMSEk[n]

E 1113954θk(nT) minus θk(nT)11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

1113970

(49)

RMSET

1KNT

1113944

K

k11113944

NT


11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(50)















20 30 40 50 6010Time index n


DoA

(deg)

0

30

60

90

120

150

180












PF[]RMSET

[deg]PF[]

RMSET

[deg]PF[]

RMSET

[deg]


RMSE3[n]



06

12

RMSE

(deg )

RMSE2[n]


06

12

RMSE

(deg )

RMSE1[n]


06

12

RMSE

(deg )






f (θ)

40 60 80 100 120 140 16020θ (deg)

0

5

10

15

20






12 097 441 05080 097 2216 11589 097 0 0378716 097 004 02791 095 031 04360 097 0 0313520 095 0 01884 093 0 02937 096 0 02433















0408

RMSE

(deg) RMSE5[n]


0408

RMSE

(deg) RMSE4[n]


0408

RMSE

(deg) RMSE3[n]


0408

RMSE

(deg) RMSE2[n]


0408

RMSE

(deg) RMSE1[n]




Ns 200

8 Conclusion


Appendix





1113960 1113961 minus N0IN (A1)




rm(t) 1113944K

k1pke


(A2)



by


(A3)



K (t)1113872 1113873 (A4)




1113957rm 1m

1113944

m

k11113944

Ns

l1


1 minus ρNs



1113957rm 1

M1113944

M

k11113944

Ns

l1


1 minus ρNs







K1113872 1113873 (A7)



E cTek[n]11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877 E cTVΛnVminus 1ek[0]+ 1113944

n

m1cTVΛnminus m


111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2


k [0]1113960 1113961 VΛnVminus 11113872 1113873

Tc

+ 1113944n

m1cTVΛnminus m


(B1)


ek[0] 1113954yk[0 | 0] minus yk(0)

1113957θk[0 | 0] minus θk(0)



T



T2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B2)

εk[0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B3)



E ek[0]eTk [0]1113960 1113961

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B4)


Data Availability




Acknowledgments






References











































where

F

1 T 05T2

0 1 T

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (5)



11138681113868111386811138681113868 (6)










a

b

c

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p 1113936minus 1

mminus p

m 1113936minus 1

mminus p

m2

1113936minus 1

mminus p

m 1113936minus 1

mminus p

m2 1113936minus 1

mminus p

m3

1113936minus 1

mminus p

m2 1113936minus 1

mminus p

m3 1113936minus 1

mminus p

m4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 1

middot

1113936nminus 1

mnminus p

1113954θk[m | m]

1113936nminus 1

mnminus p


1113936nminus 1

mnminus p

(m minus n)21113954θk[m | m]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(8)


1113954yk[n | n]

c

b

2a

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (9)














fPrime(θ) θ minus

Re aprimeH(θ)Πa(θ)1113876 1113877


(12)




dH[n] 1113954Π[n]d[n] (13)



ak[n] a 1113954θk[n | n minus 1]1113872 1113873 (14)

and

dk[n] da(θ)

dθ

111386811138681113868111386811138681113868 θ1113954θk[n | nminus 1] (15)










1 minus ρNs


1 minus ρNs


1 minus ρNs



(16)




11 minus α





(19)




(20)



Ns minus Ns + 1( 1113857ρ + ρNs+11113858 1113859


(21)















(25)



(26)





(27)










RMSEk

1NT

1113944

NT


11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(30)


RMSEk

1NT

1113944

NT

n1E cTek[n]

11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(31)



cTek[n](32)


Lemma 1










gk1 1 minus κ1κ22

gk2 12T


gk3 1


(36)




λ3 + gk1 + gk2T + 05gk3T2

minus 31113872 1113873λ2


+ 31113872 1113873λ + gk1 minus 1 0(37)


λ3 + gk1 + gk2T + 05gk3T2

minus 31113872 1113873λ2


+ 31113872 1113873λ + gk1 minus 1


(38)





(39)



m1VΛnminus m


+ 1113944n

m1VΛnminus m


1113960 1113961δk[m]

(40)





m1VΛnminus m


(41)



E εk[n]1113858 1113859 0 (42)

and that

E εk[m]εk[n]1113858 1113859 δmnσ2k[n] (43)

en

RMSEk

1NT

1113944

NT


1113872 1113873Tc + 1113944

n

m1cTVΛnminus m


11139741113972

(44)

where

T

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(45)




σk[n] 1113957σk (46)

for all n then

RMSEk 1113957σk

1NT

1113944

NT


1113872 1113873Tc + 1113944

n

m1cTVΛnminus m

(I minus Λ)Vminus 1c1113872 11138732

⎡⎣ ⎤⎦

11139741113972

(47)

where

T

11T

1T2

1T

2T2

3T3

1T2

3T3

6T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(48)


(1NT) 1113936NT

n σ2k[n]

1113969










(1NT) 1113936NT

n1 σ2k[n]

1113969






02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

088

091

087 088 091

152

088087

087088 09


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

091

152

5

1

215

09

1


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

09

088

1

215

1

09

1


oA (deg

)

20 30 40 50 6010Time index n

0

30

60

90

120

150

180


θ4 (nT)θ5 (nT)





20 30 40 50 60100Time index n

σ12[n]

σ22[n]

σ32[n]

σ42[n]

σ52[n]

σ2 k[n]

0

05

1

15

2

25



σ21[n] σ22[n] σ23[n] σ24[n] σ25[n]

ϕ 15deg (0965 1141) (0973 1022) (0842 0984) (0908 0980) (0799 0987)

ϕ 35deg (0965 1124) (0973 1017) (0859 0984) (0916 0980) (0822 0987)

ϕ 55deg (0967 1050) (0973 0995) (0922 0984) (0949 0980) (0908 0987)

02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09 098

095 085

08

098

09509

085

098

09509


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

097

097

0971

1

1

105

105

11






RMSEk[n]

E 1113954θk(nT) minus θk(nT)11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

1113970

(49)

RMSET

1KNT

1113944

K

k11113944

NT


11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(50)















20 30 40 50 6010Time index n


DoA

(deg)

0

30

60

90

120

150

180












PF[]RMSET

[deg]PF[]

RMSET

[deg]PF[]

RMSET

[deg]


RMSE3[n]



06

12

RMSE

(deg )

RMSE2[n]


06

12

RMSE

(deg )

RMSE1[n]


06

12

RMSE

(deg )






f (θ)

40 60 80 100 120 140 16020θ (deg)

0

5

10

15

20






12 097 441 05080 097 2216 11589 097 0 0378716 097 004 02791 095 031 04360 097 0 0313520 095 0 01884 093 0 02937 096 0 02433















0408

RMSE

(deg) RMSE5[n]


0408

RMSE

(deg) RMSE4[n]


0408

RMSE

(deg) RMSE3[n]


0408

RMSE

(deg) RMSE2[n]


0408

RMSE

(deg) RMSE1[n]




Ns 200

8 Conclusion


Appendix





1113960 1113961 minus N0IN (A1)




rm(t) 1113944K

k1pke


(A2)



by


(A3)



K (t)1113872 1113873 (A4)




1113957rm 1m

1113944

m

k11113944

Ns

l1


1 minus ρNs



1113957rm 1

M1113944

M

k11113944

Ns

l1


1 minus ρNs







K1113872 1113873 (A7)



E cTek[n]11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877 E cTVΛnVminus 1ek[0]+ 1113944

n

m1cTVΛnminus m


111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2


k [0]1113960 1113961 VΛnVminus 11113872 1113873

Tc

+ 1113944n

m1cTVΛnminus m


(B1)


ek[0] 1113954yk[0 | 0] minus yk(0)

1113957θk[0 | 0] minus θk(0)



T



T2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B2)

εk[0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B3)



E ek[0]eTk [0]1113960 1113961

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B4)


Data Availability




Acknowledgments






References


































a

b

c

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p 1113936minus 1

mminus p

m 1113936minus 1

mminus p

m2

1113936minus 1

mminus p

m 1113936minus 1

mminus p

m2 1113936minus 1

mminus p

m3

1113936minus 1

mminus p

m2 1113936minus 1

mminus p

m3 1113936minus 1

mminus p

m4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 1

middot

1113936nminus 1

mnminus p

1113954θk[m | m]

1113936nminus 1

mnminus p


1113936nminus 1

mnminus p

(m minus n)21113954θk[m | m]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(8)


1113954yk[n | n]

c

b

2a

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (9)














fPrime(θ) θ minus

Re aprimeH(θ)Πa(θ)1113876 1113877


(12)




dH[n] 1113954Π[n]d[n] (13)



ak[n] a 1113954θk[n | n minus 1]1113872 1113873 (14)

and

dk[n] da(θ)

dθ

111386811138681113868111386811138681113868 θ1113954θk[n | nminus 1] (15)










1 minus ρNs


1 minus ρNs


1 minus ρNs



(16)




11 minus α





(19)




(20)



Ns minus Ns + 1( 1113857ρ + ρNs+11113858 1113859


(21)















(25)



(26)





(27)










RMSEk

1NT

1113944

NT


11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(30)


RMSEk

1NT

1113944

NT

n1E cTek[n]

11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(31)



cTek[n](32)


Lemma 1










gk1 1 minus κ1κ22

gk2 12T


gk3 1


(36)




λ3 + gk1 + gk2T + 05gk3T2

minus 31113872 1113873λ2


+ 31113872 1113873λ + gk1 minus 1 0(37)


λ3 + gk1 + gk2T + 05gk3T2

minus 31113872 1113873λ2


+ 31113872 1113873λ + gk1 minus 1


(38)





(39)



m1VΛnminus m


+ 1113944n

m1VΛnminus m


1113960 1113961δk[m]

(40)





m1VΛnminus m


(41)



E εk[n]1113858 1113859 0 (42)

and that

E εk[m]εk[n]1113858 1113859 δmnσ2k[n] (43)

en

RMSEk

1NT

1113944

NT


1113872 1113873Tc + 1113944

n

m1cTVΛnminus m


11139741113972

(44)

where

T

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(45)




σk[n] 1113957σk (46)

for all n then

RMSEk 1113957σk

1NT

1113944

NT


1113872 1113873Tc + 1113944

n

m1cTVΛnminus m

(I minus Λ)Vminus 1c1113872 11138732

⎡⎣ ⎤⎦

11139741113972

(47)

where

T

11T

1T2

1T

2T2

3T3

1T2

3T3

6T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(48)


(1NT) 1113936NT

n σ2k[n]

1113969










(1NT) 1113936NT

n1 σ2k[n]

1113969






02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

088

091

087 088 091

152

088087

087088 09


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

091

152

5

1

215

09

1


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

09

088

1

215

1

09

1


oA (deg

)

20 30 40 50 6010Time index n

0

30

60

90

120

150

180


θ4 (nT)θ5 (nT)





20 30 40 50 60100Time index n

σ12[n]

σ22[n]

σ32[n]

σ42[n]

σ52[n]

σ2 k[n]

0

05

1

15

2

25



σ21[n] σ22[n] σ23[n] σ24[n] σ25[n]

ϕ 15deg (0965 1141) (0973 1022) (0842 0984) (0908 0980) (0799 0987)

ϕ 35deg (0965 1124) (0973 1017) (0859 0984) (0916 0980) (0822 0987)

ϕ 55deg (0967 1050) (0973 0995) (0922 0984) (0949 0980) (0908 0987)

02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09 098

095 085

08

098

09509

085

098

09509


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

097

097

0971

1

1

105

105

11






RMSEk[n]

E 1113954θk(nT) minus θk(nT)11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

1113970

(49)

RMSET

1KNT

1113944

K

k11113944

NT


11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(50)















20 30 40 50 6010Time index n


DoA

(deg)

0

30

60

90

120

150

180












PF[]RMSET

[deg]PF[]

RMSET

[deg]PF[]

RMSET

[deg]


RMSE3[n]



06

12

RMSE

(deg )

RMSE2[n]


06

12

RMSE

(deg )

RMSE1[n]


06

12

RMSE

(deg )






f (θ)

40 60 80 100 120 140 16020θ (deg)

0

5

10

15

20






12 097 441 05080 097 2216 11589 097 0 0378716 097 004 02791 095 031 04360 097 0 0313520 095 0 01884 093 0 02937 096 0 02433















0408

RMSE

(deg) RMSE5[n]


0408

RMSE

(deg) RMSE4[n]


0408

RMSE

(deg) RMSE3[n]


0408

RMSE

(deg) RMSE2[n]


0408

RMSE

(deg) RMSE1[n]




Ns 200

8 Conclusion


Appendix





1113960 1113961 minus N0IN (A1)




rm(t) 1113944K

k1pke


(A2)



by


(A3)



K (t)1113872 1113873 (A4)




1113957rm 1m

1113944

m

k11113944

Ns

l1


1 minus ρNs



1113957rm 1

M1113944

M

k11113944

Ns

l1


1 minus ρNs







K1113872 1113873 (A7)



E cTek[n]11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877 E cTVΛnVminus 1ek[0]+ 1113944

n

m1cTVΛnminus m


111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2


k [0]1113960 1113961 VΛnVminus 11113872 1113873

Tc

+ 1113944n

m1cTVΛnminus m


(B1)


ek[0] 1113954yk[0 | 0] minus yk(0)

1113957θk[0 | 0] minus θk(0)



T



T2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B2)

εk[0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B3)



E ek[0]eTk [0]1113960 1113961

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B4)


Data Availability




Acknowledgments






References






























ak[n] a 1113954θk[n | n minus 1]1113872 1113873 (14)

and

dk[n] da(θ)

dθ

111386811138681113868111386811138681113868 θ1113954θk[n | nminus 1] (15)










1 minus ρNs


1 minus ρNs


1 minus ρNs



(16)




11 minus α





(19)




(20)



Ns minus Ns + 1( 1113857ρ + ρNs+11113858 1113859


(21)















(25)



(26)





(27)










RMSEk

1NT

1113944

NT


11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(30)


RMSEk

1NT

1113944

NT

n1E cTek[n]

11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(31)



cTek[n](32)


Lemma 1










gk1 1 minus κ1κ22

gk2 12T


gk3 1


(36)




λ3 + gk1 + gk2T + 05gk3T2

minus 31113872 1113873λ2


+ 31113872 1113873λ + gk1 minus 1 0(37)


λ3 + gk1 + gk2T + 05gk3T2

minus 31113872 1113873λ2


+ 31113872 1113873λ + gk1 minus 1


(38)





(39)



m1VΛnminus m


+ 1113944n

m1VΛnminus m


1113960 1113961δk[m]

(40)





m1VΛnminus m


(41)



E εk[n]1113858 1113859 0 (42)

and that

E εk[m]εk[n]1113858 1113859 δmnσ2k[n] (43)

en

RMSEk

1NT

1113944

NT


1113872 1113873Tc + 1113944

n

m1cTVΛnminus m


11139741113972

(44)

where

T

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(45)




σk[n] 1113957σk (46)

for all n then

RMSEk 1113957σk

1NT

1113944

NT


1113872 1113873Tc + 1113944

n

m1cTVΛnminus m

(I minus Λ)Vminus 1c1113872 11138732

⎡⎣ ⎤⎦

11139741113972

(47)

where

T

11T

1T2

1T

2T2

3T3

1T2

3T3

6T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(48)


(1NT) 1113936NT

n σ2k[n]

1113969










(1NT) 1113936NT

n1 σ2k[n]

1113969






02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

088

091

087 088 091

152

088087

087088 09


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

091

152

5

1

215

09

1


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

09

088

1

215

1

09

1


oA (deg

)

20 30 40 50 6010Time index n

0

30

60

90

120

150

180


θ4 (nT)θ5 (nT)





20 30 40 50 60100Time index n

σ12[n]

σ22[n]

σ32[n]

σ42[n]

σ52[n]

σ2 k[n]

0

05

1

15

2

25



σ21[n] σ22[n] σ23[n] σ24[n] σ25[n]

ϕ 15deg (0965 1141) (0973 1022) (0842 0984) (0908 0980) (0799 0987)

ϕ 35deg (0965 1124) (0973 1017) (0859 0984) (0916 0980) (0822 0987)

ϕ 55deg (0967 1050) (0973 0995) (0922 0984) (0949 0980) (0908 0987)

02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09 098

095 085

08

098

09509

085

098

09509


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

097

097

0971

1

1

105

105

11






RMSEk[n]

E 1113954θk(nT) minus θk(nT)11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

1113970

(49)

RMSET

1KNT

1113944

K

k11113944

NT


11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(50)















20 30 40 50 6010Time index n


DoA

(deg)

0

30

60

90

120

150

180












PF[]RMSET

[deg]PF[]

RMSET

[deg]PF[]

RMSET

[deg]


RMSE3[n]



06

12

RMSE

(deg )

RMSE2[n]


06

12

RMSE

(deg )

RMSE1[n]


06

12

RMSE

(deg )






f (θ)

40 60 80 100 120 140 16020θ (deg)

0

5

10

15

20






12 097 441 05080 097 2216 11589 097 0 0378716 097 004 02791 095 031 04360 097 0 0313520 095 0 01884 093 0 02937 096 0 02433















0408

RMSE

(deg) RMSE5[n]


0408

RMSE

(deg) RMSE4[n]


0408

RMSE

(deg) RMSE3[n]


0408

RMSE

(deg) RMSE2[n]


0408

RMSE

(deg) RMSE1[n]




Ns 200

8 Conclusion


Appendix





1113960 1113961 minus N0IN (A1)




rm(t) 1113944K

k1pke


(A2)



by


(A3)



K (t)1113872 1113873 (A4)




1113957rm 1m

1113944

m

k11113944

Ns

l1


1 minus ρNs



1113957rm 1

M1113944

M

k11113944

Ns

l1


1 minus ρNs







K1113872 1113873 (A7)



E cTek[n]11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877 E cTVΛnVminus 1ek[0]+ 1113944

n

m1cTVΛnminus m


111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2


k [0]1113960 1113961 VΛnVminus 11113872 1113873

Tc

+ 1113944n

m1cTVΛnminus m


(B1)


ek[0] 1113954yk[0 | 0] minus yk(0)

1113957θk[0 | 0] minus θk(0)



T



T2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B2)

εk[0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B3)



E ek[0]eTk [0]1113960 1113961

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B4)


Data Availability




Acknowledgments






References






































(25)



(26)





(27)










RMSEk

1NT

1113944

NT


11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(30)


RMSEk

1NT

1113944

NT

n1E cTek[n]

11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(31)



cTek[n](32)


Lemma 1










gk1 1 minus κ1κ22

gk2 12T


gk3 1


(36)




λ3 + gk1 + gk2T + 05gk3T2

minus 31113872 1113873λ2


+ 31113872 1113873λ + gk1 minus 1 0(37)


λ3 + gk1 + gk2T + 05gk3T2

minus 31113872 1113873λ2


+ 31113872 1113873λ + gk1 minus 1


(38)





(39)



m1VΛnminus m


+ 1113944n

m1VΛnminus m


1113960 1113961δk[m]

(40)





m1VΛnminus m


(41)



E εk[n]1113858 1113859 0 (42)

and that

E εk[m]εk[n]1113858 1113859 δmnσ2k[n] (43)

en

RMSEk

1NT

1113944

NT


1113872 1113873Tc + 1113944

n

m1cTVΛnminus m


11139741113972

(44)

where

T

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(45)




σk[n] 1113957σk (46)

for all n then

RMSEk 1113957σk

1NT

1113944

NT


1113872 1113873Tc + 1113944

n

m1cTVΛnminus m

(I minus Λ)Vminus 1c1113872 11138732

⎡⎣ ⎤⎦

11139741113972

(47)

where

T

11T

1T2

1T

2T2

3T3

1T2

3T3

6T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(48)


(1NT) 1113936NT

n σ2k[n]

1113969










(1NT) 1113936NT

n1 σ2k[n]

1113969






02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

088

091

087 088 091

152

088087

087088 09


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

091

152

5

1

215

09

1


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

09

088

1

215

1

09

1


oA (deg

)

20 30 40 50 6010Time index n

0

30

60

90

120

150

180


θ4 (nT)θ5 (nT)





20 30 40 50 60100Time index n

σ12[n]

σ22[n]

σ32[n]

σ42[n]

σ52[n]

σ2 k[n]

0

05

1

15

2

25



σ21[n] σ22[n] σ23[n] σ24[n] σ25[n]

ϕ 15deg (0965 1141) (0973 1022) (0842 0984) (0908 0980) (0799 0987)

ϕ 35deg (0965 1124) (0973 1017) (0859 0984) (0916 0980) (0822 0987)

ϕ 55deg (0967 1050) (0973 0995) (0922 0984) (0949 0980) (0908 0987)

02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09 098

095 085

08

098

09509

085

098

09509


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

097

097

0971

1

1

105

105

11






RMSEk[n]

E 1113954θk(nT) minus θk(nT)11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

1113970

(49)

RMSET

1KNT

1113944

K

k11113944

NT


11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(50)















20 30 40 50 6010Time index n


DoA

(deg)

0

30

60

90

120

150

180












PF[]RMSET

[deg]PF[]

RMSET

[deg]PF[]

RMSET

[deg]


RMSE3[n]



06

12

RMSE

(deg )

RMSE2[n]


06

12

RMSE

(deg )

RMSE1[n]


06

12

RMSE

(deg )






f (θ)

40 60 80 100 120 140 16020θ (deg)

0

5

10

15

20






12 097 441 05080 097 2216 11589 097 0 0378716 097 004 02791 095 031 04360 097 0 0313520 095 0 01884 093 0 02937 096 0 02433















0408

RMSE

(deg) RMSE5[n]


0408

RMSE

(deg) RMSE4[n]


0408

RMSE

(deg) RMSE3[n]


0408

RMSE

(deg) RMSE2[n]


0408

RMSE

(deg) RMSE1[n]




Ns 200

8 Conclusion


Appendix





1113960 1113961 minus N0IN (A1)




rm(t) 1113944K

k1pke


(A2)



by


(A3)



K (t)1113872 1113873 (A4)




1113957rm 1m

1113944

m

k11113944

Ns

l1


1 minus ρNs



1113957rm 1

M1113944

M

k11113944

Ns

l1


1 minus ρNs







K1113872 1113873 (A7)



E cTek[n]11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877 E cTVΛnVminus 1ek[0]+ 1113944

n

m1cTVΛnminus m


111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2


k [0]1113960 1113961 VΛnVminus 11113872 1113873

Tc

+ 1113944n

m1cTVΛnminus m


(B1)


ek[0] 1113954yk[0 | 0] minus yk(0)

1113957θk[0 | 0] minus θk(0)



T



T2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B2)

εk[0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B3)



E ek[0]eTk [0]1113960 1113961

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B4)


Data Availability




Acknowledgments






References






























λ3 + gk1 + gk2T + 05gk3T2

minus 31113872 1113873λ2


+ 31113872 1113873λ + gk1 minus 1 0(37)


λ3 + gk1 + gk2T + 05gk3T2

minus 31113872 1113873λ2


+ 31113872 1113873λ + gk1 minus 1


(38)





(39)



m1VΛnminus m


+ 1113944n

m1VΛnminus m


1113960 1113961δk[m]

(40)





m1VΛnminus m


(41)



E εk[n]1113858 1113859 0 (42)

and that

E εk[m]εk[n]1113858 1113859 δmnσ2k[n] (43)

en

RMSEk

1NT

1113944

NT


1113872 1113873Tc + 1113944

n

m1cTVΛnminus m


11139741113972

(44)

where

T

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(45)




σk[n] 1113957σk (46)

for all n then

RMSEk 1113957σk

1NT

1113944

NT


1113872 1113873Tc + 1113944

n

m1cTVΛnminus m

(I minus Λ)Vminus 1c1113872 11138732

⎡⎣ ⎤⎦

11139741113972

(47)

where

T

11T

1T2

1T

2T2

3T3

1T2

3T3

6T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(48)


(1NT) 1113936NT

n σ2k[n]

1113969










(1NT) 1113936NT

n1 σ2k[n]

1113969






02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

088

091

087 088 091

152

088087

087088 09


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

091

152

5

1

215

09

1


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

09

088

1

215

1

09

1


oA (deg

)

20 30 40 50 6010Time index n

0

30

60

90

120

150

180


θ4 (nT)θ5 (nT)





20 30 40 50 60100Time index n

σ12[n]

σ22[n]

σ32[n]

σ42[n]

σ52[n]

σ2 k[n]

0

05

1

15

2

25



σ21[n] σ22[n] σ23[n] σ24[n] σ25[n]

ϕ 15deg (0965 1141) (0973 1022) (0842 0984) (0908 0980) (0799 0987)

ϕ 35deg (0965 1124) (0973 1017) (0859 0984) (0916 0980) (0822 0987)

ϕ 55deg (0967 1050) (0973 0995) (0922 0984) (0949 0980) (0908 0987)

02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09 098

095 085

08

098

09509

085

098

09509


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

097

097

0971

1

1

105

105

11






RMSEk[n]

E 1113954θk(nT) minus θk(nT)11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

1113970

(49)

RMSET

1KNT

1113944

K

k11113944

NT


11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(50)















20 30 40 50 6010Time index n


DoA

(deg)

0

30

60

90

120

150

180












PF[]RMSET

[deg]PF[]

RMSET

[deg]PF[]

RMSET

[deg]


RMSE3[n]



06

12

RMSE

(deg )

RMSE2[n]


06

12

RMSE

(deg )

RMSE1[n]


06

12

RMSE

(deg )






f (θ)

40 60 80 100 120 140 16020θ (deg)

0

5

10

15

20






12 097 441 05080 097 2216 11589 097 0 0378716 097 004 02791 095 031 04360 097 0 0313520 095 0 01884 093 0 02937 096 0 02433















0408

RMSE

(deg) RMSE5[n]


0408

RMSE

(deg) RMSE4[n]


0408

RMSE

(deg) RMSE3[n]


0408

RMSE

(deg) RMSE2[n]


0408

RMSE

(deg) RMSE1[n]




Ns 200

8 Conclusion


Appendix





1113960 1113961 minus N0IN (A1)




rm(t) 1113944K

k1pke


(A2)



by


(A3)



K (t)1113872 1113873 (A4)




1113957rm 1m

1113944

m

k11113944

Ns

l1


1 minus ρNs



1113957rm 1

M1113944

M

k11113944

Ns

l1


1 minus ρNs







K1113872 1113873 (A7)



E cTek[n]11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877 E cTVΛnVminus 1ek[0]+ 1113944

n

m1cTVΛnminus m


111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2


k [0]1113960 1113961 VΛnVminus 11113872 1113873

Tc

+ 1113944n

m1cTVΛnminus m


(B1)


ek[0] 1113954yk[0 | 0] minus yk(0)

1113957θk[0 | 0] minus θk(0)



T



T2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B2)

εk[0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B3)



E ek[0]eTk [0]1113960 1113961

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B4)


Data Availability




Acknowledgments






References
































σk[n] 1113957σk (46)

for all n then

RMSEk 1113957σk

1NT

1113944

NT


1113872 1113873Tc + 1113944

n

m1cTVΛnminus m

(I minus Λ)Vminus 1c1113872 11138732

⎡⎣ ⎤⎦

11139741113972

(47)

where

T

11T

1T2

1T

2T2

3T3

1T2

3T3

6T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(48)


(1NT) 1113936NT

n σ2k[n]

1113969










(1NT) 1113936NT

n1 σ2k[n]

1113969






02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

088

091

087 088 091

152

088087

087088 09


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

091

152

5

1

215

09

1


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

09

088

1

215

1

09

1


oA (deg

)

20 30 40 50 6010Time index n

0

30

60

90

120

150

180


θ4 (nT)θ5 (nT)





20 30 40 50 60100Time index n

σ12[n]

σ22[n]

σ32[n]

σ42[n]

σ52[n]

σ2 k[n]

0

05

1

15

2

25



σ21[n] σ22[n] σ23[n] σ24[n] σ25[n]

ϕ 15deg (0965 1141) (0973 1022) (0842 0984) (0908 0980) (0799 0987)

ϕ 35deg (0965 1124) (0973 1017) (0859 0984) (0916 0980) (0822 0987)

ϕ 55deg (0967 1050) (0973 0995) (0922 0984) (0949 0980) (0908 0987)

02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09 098

095 085

08

098

09509

085

098

09509


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

097

097

0971

1

1

105

105

11






RMSEk[n]

E 1113954θk(nT) minus θk(nT)11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

1113970

(49)

RMSET

1KNT

1113944

K

k11113944

NT


11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(50)















20 30 40 50 6010Time index n


DoA

(deg)

0

30

60

90

120

150

180












PF[]RMSET

[deg]PF[]

RMSET

[deg]PF[]

RMSET

[deg]


RMSE3[n]



06

12

RMSE

(deg )

RMSE2[n]


06

12

RMSE

(deg )

RMSE1[n]


06

12

RMSE

(deg )






f (θ)

40 60 80 100 120 140 16020θ (deg)

0

5

10

15

20






12 097 441 05080 097 2216 11589 097 0 0378716 097 004 02791 095 031 04360 097 0 0313520 095 0 01884 093 0 02937 096 0 02433















0408

RMSE

(deg) RMSE5[n]


0408

RMSE

(deg) RMSE4[n]


0408

RMSE

(deg) RMSE3[n]


0408

RMSE

(deg) RMSE2[n]


0408

RMSE

(deg) RMSE1[n]




Ns 200

8 Conclusion


Appendix





1113960 1113961 minus N0IN (A1)




rm(t) 1113944K

k1pke


(A2)



by


(A3)



K (t)1113872 1113873 (A4)




1113957rm 1m

1113944

m

k11113944

Ns

l1


1 minus ρNs



1113957rm 1

M1113944

M

k11113944

Ns

l1


1 minus ρNs







K1113872 1113873 (A7)



E cTek[n]11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877 E cTVΛnVminus 1ek[0]+ 1113944

n

m1cTVΛnminus m


111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2


k [0]1113960 1113961 VΛnVminus 11113872 1113873

Tc

+ 1113944n

m1cTVΛnminus m


(B1)


ek[0] 1113954yk[0 | 0] minus yk(0)

1113957θk[0 | 0] minus θk(0)



T



T2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B2)

εk[0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B3)



E ek[0]eTk [0]1113960 1113961

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B4)


Data Availability




Acknowledgments






References

































02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

088

091

087 088 091

152

088087

087088 09


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

091

152

5

1

215

09

1


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

09

09

09

088

1

215

1

09

1


oA (deg

)

20 30 40 50 6010Time index n

0

30

60

90

120

150

180


θ4 (nT)θ5 (nT)





20 30 40 50 60100Time index n

σ12[n]

σ22[n]

σ32[n]

σ42[n]

σ52[n]

σ2 k[n]

0

05

1

15

2

25



σ21[n] σ22[n] σ23[n] σ24[n] σ25[n]

ϕ 15deg (0965 1141) (0973 1022) (0842 0984) (0908 0980) (0799 0987)

ϕ 35deg (0965 1124) (0973 1017) (0859 0984) (0916 0980) (0822 0987)

ϕ 55deg (0967 1050) (0973 0995) (0922 0984) (0949 0980) (0908 0987)

02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09 098

095 085

08

098

09509

085

098

09509


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

097

097

0971

1

1

105

105

11






RMSEk[n]

E 1113954θk(nT) minus θk(nT)11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

1113970

(49)

RMSET

1KNT

1113944

K

k11113944

NT


11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(50)















20 30 40 50 6010Time index n


DoA

(deg)

0

30

60

90

120

150

180












PF[]RMSET

[deg]PF[]

RMSET

[deg]PF[]

RMSET

[deg]


RMSE3[n]



06

12

RMSE

(deg )

RMSE2[n]


06

12

RMSE

(deg )

RMSE1[n]


06

12

RMSE

(deg )






f (θ)

40 60 80 100 120 140 16020θ (deg)

0

5

10

15

20






12 097 441 05080 097 2216 11589 097 0 0378716 097 004 02791 095 031 04360 097 0 0313520 095 0 01884 093 0 02937 096 0 02433















0408

RMSE

(deg) RMSE5[n]


0408

RMSE

(deg) RMSE4[n]


0408

RMSE

(deg) RMSE3[n]


0408

RMSE

(deg) RMSE2[n]


0408

RMSE

(deg) RMSE1[n]




Ns 200

8 Conclusion


Appendix





1113960 1113961 minus N0IN (A1)




rm(t) 1113944K

k1pke


(A2)



by


(A3)



K (t)1113872 1113873 (A4)




1113957rm 1m

1113944

m

k11113944

Ns

l1


1 minus ρNs



1113957rm 1

M1113944

M

k11113944

Ns

l1


1 minus ρNs







K1113872 1113873 (A7)



E cTek[n]11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877 E cTVΛnVminus 1ek[0]+ 1113944

n

m1cTVΛnminus m


111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2


k [0]1113960 1113961 VΛnVminus 11113872 1113873

Tc

+ 1113944n

m1cTVΛnminus m


(B1)


ek[0] 1113954yk[0 | 0] minus yk(0)

1113957θk[0 | 0] minus θk(0)



T



T2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B2)

εk[0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B3)



E ek[0]eTk [0]1113960 1113961

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B4)


Data Availability




Acknowledgments






References
































20 30 40 50 60100Time index n

σ12[n]

σ22[n]

σ32[n]

σ42[n]

σ52[n]

σ2 k[n]

0

05

1

15

2

25



σ21[n] σ22[n] σ23[n] σ24[n] σ25[n]

ϕ 15deg (0965 1141) (0973 1022) (0842 0984) (0908 0980) (0799 0987)

ϕ 35deg (0965 1124) (0973 1017) (0859 0984) (0916 0980) (0822 0987)

ϕ 55deg (0967 1050) (0973 0995) (0922 0984) (0949 0980) (0908 0987)

02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09 098

095 085

08

098

09509

085

098

09509


02 03 04 05 06 07 08 0901κ1

κ2

01

02

03

04

05

06

07

08

09

097

097

0971

1

1

105

105

11






RMSEk[n]

E 1113954θk(nT) minus θk(nT)11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

1113970

(49)

RMSET

1KNT

1113944

K

k11113944

NT


11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(50)















20 30 40 50 6010Time index n


DoA

(deg)

0

30

60

90

120

150

180












PF[]RMSET

[deg]PF[]

RMSET

[deg]PF[]

RMSET

[deg]


RMSE3[n]



06

12

RMSE

(deg )

RMSE2[n]


06

12

RMSE

(deg )

RMSE1[n]


06

12

RMSE

(deg )






f (θ)

40 60 80 100 120 140 16020θ (deg)

0

5

10

15

20






12 097 441 05080 097 2216 11589 097 0 0378716 097 004 02791 095 031 04360 097 0 0313520 095 0 01884 093 0 02937 096 0 02433















0408

RMSE

(deg) RMSE5[n]


0408

RMSE

(deg) RMSE4[n]


0408

RMSE

(deg) RMSE3[n]


0408

RMSE

(deg) RMSE2[n]


0408

RMSE

(deg) RMSE1[n]




Ns 200

8 Conclusion


Appendix





1113960 1113961 minus N0IN (A1)




rm(t) 1113944K

k1pke


(A2)



by


(A3)



K (t)1113872 1113873 (A4)




1113957rm 1m

1113944

m

k11113944

Ns

l1


1 minus ρNs



1113957rm 1

M1113944

M

k11113944

Ns

l1


1 minus ρNs







K1113872 1113873 (A7)



E cTek[n]11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877 E cTVΛnVminus 1ek[0]+ 1113944

n

m1cTVΛnminus m


111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2


k [0]1113960 1113961 VΛnVminus 11113872 1113873

Tc

+ 1113944n

m1cTVΛnminus m


(B1)


ek[0] 1113954yk[0 | 0] minus yk(0)

1113957θk[0 | 0] minus θk(0)



T



T2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B2)

εk[0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B3)



E ek[0]eTk [0]1113960 1113961

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B4)


Data Availability




Acknowledgments






References

































RMSEk[n]

E 1113954θk(nT) minus θk(nT)11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

1113970

(49)

RMSET

1KNT

1113944

K

k11113944

NT


11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

11139741113972

(50)















20 30 40 50 6010Time index n


DoA

(deg)

0

30

60

90

120

150

180












PF[]RMSET

[deg]PF[]

RMSET

[deg]PF[]

RMSET

[deg]


RMSE3[n]



06

12

RMSE

(deg )

RMSE2[n]


06

12

RMSE

(deg )

RMSE1[n]


06

12

RMSE

(deg )






f (θ)

40 60 80 100 120 140 16020θ (deg)

0

5

10

15

20






12 097 441 05080 097 2216 11589 097 0 0378716 097 004 02791 095 031 04360 097 0 0313520 095 0 01884 093 0 02937 096 0 02433















0408

RMSE

(deg) RMSE5[n]


0408

RMSE

(deg) RMSE4[n]


0408

RMSE

(deg) RMSE3[n]


0408

RMSE

(deg) RMSE2[n]


0408

RMSE

(deg) RMSE1[n]




Ns 200

8 Conclusion


Appendix





1113960 1113961 minus N0IN (A1)




rm(t) 1113944K

k1pke


(A2)



by


(A3)



K (t)1113872 1113873 (A4)




1113957rm 1m

1113944

m

k11113944

Ns

l1


1 minus ρNs



1113957rm 1

M1113944

M

k11113944

Ns

l1


1 minus ρNs







K1113872 1113873 (A7)



E cTek[n]11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877 E cTVΛnVminus 1ek[0]+ 1113944

n

m1cTVΛnminus m


111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2


k [0]1113960 1113961 VΛnVminus 11113872 1113873

Tc

+ 1113944n

m1cTVΛnminus m


(B1)


ek[0] 1113954yk[0 | 0] minus yk(0)

1113957θk[0 | 0] minus θk(0)



T



T2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B2)

εk[0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B3)



E ek[0]eTk [0]1113960 1113961

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B4)


Data Availability




Acknowledgments






References







































PF[]RMSET

[deg]PF[]

RMSET

[deg]PF[]

RMSET

[deg]


RMSE3[n]



06

12

RMSE

(deg )

RMSE2[n]


06

12

RMSE

(deg )

RMSE1[n]


06

12

RMSE

(deg )






f (θ)

40 60 80 100 120 140 16020θ (deg)

0

5

10

15

20






12 097 441 05080 097 2216 11589 097 0 0378716 097 004 02791 095 031 04360 097 0 0313520 095 0 01884 093 0 02937 096 0 02433















0408

RMSE

(deg) RMSE5[n]


0408

RMSE

(deg) RMSE4[n]


0408

RMSE

(deg) RMSE3[n]


0408

RMSE

(deg) RMSE2[n]


0408

RMSE

(deg) RMSE1[n]




Ns 200

8 Conclusion


Appendix





1113960 1113961 minus N0IN (A1)




rm(t) 1113944K

k1pke


(A2)



by


(A3)



K (t)1113872 1113873 (A4)




1113957rm 1m

1113944

m

k11113944

Ns

l1


1 minus ρNs



1113957rm 1

M1113944

M

k11113944

Ns

l1


1 minus ρNs







K1113872 1113873 (A7)



E cTek[n]11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877 E cTVΛnVminus 1ek[0]+ 1113944

n

m1cTVΛnminus m


111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2


k [0]1113960 1113961 VΛnVminus 11113872 1113873

Tc

+ 1113944n

m1cTVΛnminus m


(B1)


ek[0] 1113954yk[0 | 0] minus yk(0)

1113957θk[0 | 0] minus θk(0)



T



T2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B2)

εk[0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B3)



E ek[0]eTk [0]1113960 1113961

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B4)


Data Availability




Acknowledgments






References

































f (θ)

40 60 80 100 120 140 16020θ (deg)

0

5

10

15

20






12 097 441 05080 097 2216 11589 097 0 0378716 097 004 02791 095 031 04360 097 0 0313520 095 0 01884 093 0 02937 096 0 02433















0408

RMSE

(deg) RMSE5[n]


0408

RMSE

(deg) RMSE4[n]


0408

RMSE

(deg) RMSE3[n]


0408

RMSE

(deg) RMSE2[n]


0408

RMSE

(deg) RMSE1[n]




Ns 200

8 Conclusion


Appendix





1113960 1113961 minus N0IN (A1)




rm(t) 1113944K

k1pke


(A2)



by


(A3)



K (t)1113872 1113873 (A4)




1113957rm 1m

1113944

m

k11113944

Ns

l1


1 minus ρNs



1113957rm 1

M1113944

M

k11113944

Ns

l1


1 minus ρNs







K1113872 1113873 (A7)



E cTek[n]11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877 E cTVΛnVminus 1ek[0]+ 1113944

n

m1cTVΛnminus m


111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2


k [0]1113960 1113961 VΛnVminus 11113872 1113873

Tc

+ 1113944n

m1cTVΛnminus m


(B1)


ek[0] 1113954yk[0 | 0] minus yk(0)

1113957θk[0 | 0] minus θk(0)



T



T2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B2)

εk[0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B3)



E ek[0]eTk [0]1113960 1113961

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B4)


Data Availability




Acknowledgments






References











































0408

RMSE

(deg) RMSE5[n]


0408

RMSE

(deg) RMSE4[n]


0408

RMSE

(deg) RMSE3[n]


0408

RMSE

(deg) RMSE2[n]


0408

RMSE

(deg) RMSE1[n]




Ns 200

8 Conclusion


Appendix





1113960 1113961 minus N0IN (A1)




rm(t) 1113944K

k1pke


(A2)



by


(A3)



K (t)1113872 1113873 (A4)




1113957rm 1m

1113944

m

k11113944

Ns

l1


1 minus ρNs



1113957rm 1

M1113944

M

k11113944

Ns

l1


1 minus ρNs







K1113872 1113873 (A7)



E cTek[n]11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877 E cTVΛnVminus 1ek[0]+ 1113944

n

m1cTVΛnminus m


111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2


k [0]1113960 1113961 VΛnVminus 11113872 1113873

Tc

+ 1113944n

m1cTVΛnminus m


(B1)


ek[0] 1113954yk[0 | 0] minus yk(0)

1113957θk[0 | 0] minus θk(0)



T



T2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B2)

εk[0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B3)



E ek[0]eTk [0]1113960 1113961

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B4)


Data Availability




Acknowledgments






References































Ns 200

8 Conclusion


Appendix





1113960 1113961 minus N0IN (A1)




rm(t) 1113944K

k1pke


(A2)



by


(A3)



K (t)1113872 1113873 (A4)




1113957rm 1m

1113944

m

k11113944

Ns

l1


1 minus ρNs



1113957rm 1

M1113944

M

k11113944

Ns

l1


1 minus ρNs







K1113872 1113873 (A7)



E cTek[n]11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877 E cTVΛnVminus 1ek[0]+ 1113944

n

m1cTVΛnminus m


111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2


k [0]1113960 1113961 VΛnVminus 11113872 1113873

Tc

+ 1113944n

m1cTVΛnminus m


(B1)


ek[0] 1113954yk[0 | 0] minus yk(0)

1113957θk[0 | 0] minus θk(0)



T



T2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B2)

εk[0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B3)



E ek[0]eTk [0]1113960 1113961

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B4)


Data Availability




Acknowledgments






References

































K1113872 1113873 (A7)



E cTek[n]11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877 E cTVΛnVminus 1ek[0]+ 1113944

n

m1cTVΛnminus m


111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2


k [0]1113960 1113961 VΛnVminus 11113872 1113873

Tc

+ 1113944n

m1cTVΛnminus m


(B1)


ek[0] 1113954yk[0 | 0] minus yk(0)

1113957θk[0 | 0] minus θk(0)



T



T2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B2)

εk[0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B3)



E ek[0]eTk [0]1113960 1113961

σ2k[0]σ2k[0]

Tσ2k[0]

σ2k[0]

T



T3

σ2k[0]



T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B4)


Data Availability




Acknowledgments






References































References






























directsignalspaceconstructionluenbergertrackerwith...

Documents