minimum distance criterion based on revised 4th august...

IET Radar, Sonar & Navigation

Research Article

DOA estimation for coprime EMVS arrays viaminimum distance criterion based onPARAFAC analysis

ISSN 1751-8784Received on 7th May 2018Revised 4th August 2018Accepted on 16th August 2018E-First on 31st October 2018doi: 10.1049/iet-rsn.2018.5155www.ietdl.org

Tanveer Ahmed1 , Zhang Xiaofei1,2,3, Zheng Wang1

1College of Electronics and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People's Republic ofChina2State Key Laboratory of Millimetre Waves, Southeast University, Nanjing 210096, People's Republic of China3National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, People's Republic of China

E-mail: [email protected]

Abstract: This study presents a minimum distance-based direction-of-arrival (DOA) estimation algorithm for coprimeelectromagnetic vector sensor (EMVS) arrays. The idea is to split-up the coprime array into two uniform linear arrays (ULAs) ofvector sensors and arrange the received ULA data in the form of a three-way array suitable for parallel factor (PARAFAC)analysis, which fits least-square models to the received source signal mixtures of ULAs and thus enables to retrieve the modelmatrices corresponding to each ULA. Nevertheless, because of the array splitting the estimated DOAs from these matrices arenot unique. To uniquely determine the DOA, the authors state and prove a theorem which is fundamental to the proposedalgorithm and provides a means to find an estimate based on the minimum distance criterion. Efficacy of the proposed algorithmis demonstrated through performance comparison with other existing algorithms such as Estimation of Signal Parameters viaRotational Invariance Technique (ESPRIT), long vector MUltiple SIgnal Classification (MUSIC), conventional PARAFAC and thepropagator method being simulated for an equivalent element ULA of EMVS and spaced half a wavelength apart. Numericalsimulations reveal that the proposed algorithm outperforms the others.

1 IntroductionDirection-of-arrival (DOA) estimation is an interesting researchproblem in array signal processing and gaining a remarkablescholarly attention due to its vast applications in various scientificand engineering disciplines including radar [1], sonar, navigation,wireless communications [2–4] radio astronomy and seismology.While traditional (scalar) sensors for DOA estimation have widelybeen explored and well understood, the use of vector sensorsprovides additional insights and offer new potential applicationsand fresh challenges. As the name implies, vector sensors measurevector quantities and more specifically an electromagnetic vectorsensor (EMVS) measures both electric and magnetic field vectorsat a point in space [5]. It is a complex antenna embodying sixspatially co-located sub-antennas such that three of them act likeshort orthogonal dipoles and sense the electrical field componentswhile the three orthogonal loop antennas sense the magnetic fieldcomponents. In addition to being sensitive to DOA, an EMVSunlike its scalar counterpart is sensitive to the polarisation ofincoming electromagnetic waves [5].

From the time when Li [6] and Nehorai and Paldi [5], coinedthe term ‘EMVS’ and its associated measuring capabilities, avariety of studies in relation with EMVS array processing havebeen comprehensively investigated. For instance, Nehorai andPaldi [5] exploited the vector cross-product of electric andmagnetic field vectors to unambiguously estimate the DOA of thesource. Studies of subspace methods such as ESPRIT [7] forarbitrarily spaced loops and dipoles to estimate direction andpolarisation were pioneered by Li [6]. Afterwards, Wong andZoltowski in [8, 9] creatively combined the method of the vectorcross-product with subspace techniques and proposed severalDOAs and polarisation estimation methods [10–13]. To cater forthe computational load offered by eigenvalue decomposition insubspace-based methods, He and Liu [13] developed acomputationally simpler direction finding an algorithm for EMVSarrays based on the propagator method [14]. According toSidiropoulos et al. [15], the idea of ESPRIT proved to have arevolutionary impact on sensor array signal processing.

Remarkably, a generic principle underlying ESPRIT hasindependently thrived in cross-disciplines, where it is usuallyadapted in a variety of ways including, trilinear/canonicaldecomposition, parallel proportional profiles and parallel factor(PARAFAC) analysis. Moreover, the structural output of EMVSnaturally leads to process the signals in multi-dimensions [15–19].For example, Sidiropoulos et al. [15] relate multiple invariancesensor array processing (MI-SAP) to PARAFAC analysis, whichessentially provides a low-rank decomposition of third or higher-order matrices (also called n-way arrays/data or tensors).Sidiropoulos et al. in [15] showed that this MI-SAP and PARAFAClinkage aids in deriving convincing results for the identifiability ofMI-SAP and proves that the uniqueness of single/MI ESPRIT isdirectly related with the uniqueness of low-rank decomposition ofthree-way arrays [16, 20]. Thus, according to Sidiropoulos et al.[15], in the context of signal processing, generalising the conceptsof ESPRIT and joint approximate diagonalisation is essentially aPARAFAC analysis.

Recent years have also seen significant scholarly efforts tohandle underdetermined DOA estimation problem [21], especiallysparse array geometries [21–24] and the associated signalprocessing algorithms [25–28], for enhanced degrees of freedomare active research topics. In a contribution by [18], Han andNehorai provided a tensorial version of data for a nested vectorsensor array and used higher-order singular value decomposition toobtain a spatially smooth covariance tensor leading to thedevelopment of tensor MUSIC algorithm. However, spatialsmoothing reduces the available degrees of freedom [22], andsecondly the computational cost associated with spectrum search inspatially smooth tensor MUSIC is also very high.

A coprime array (CPA) [22] also belongs to the class of sparseand non-uniform arrays, and in essence a linear CPA can be viewedas a superposition of two uniform linear sub-arrays when alignedalong two parallel lines as shown in Fig. 1. Thus, splitting up aCPA into two uniform linear sub-arrays, allows us to individuallyprocess each linear sub-array and then intelligently combine theresults so as to get the estimation results from the actual CPA. Inthe literature, there exist several methods for processing a CPA of

IET Radar Sonar Navig., 2019, Vol. 13 Iss. 1, pp. 65-73© The Institution of Engineering and Technology 2018

65

scalar sensors, e.g. spatially smooth MUSIC algorithm [22, 26] andcompressive sensing methods [23, 28]; however, as far as ourknowledge is concerned no similar attempt has been madepreviously from the viewpoint of estimation of DOA via coprimeEMVS array. Therefore, our goal in this paper is to solve the DOAestimation problem via coprime EMVS array by proposing aminimum distance PARAFAC algorithm.

When a CPA splits up into two uniform linear arrays (ULAs),the spacing between the adjacent vector sensors becomes largerthan half of the wavelength. In scalar sensor array, this largespacing will cause ambiguities in phase and hence estimating DOAfor one source will become difficult [29]. Since a single EMVSembodies in itself a six-element co-located sub-array and the singlevector sensor manifold is independent of these spatial phase factors[8, 9], the DOA from this coprime EMVS array can be estimatedwithout any ambiguity. Nevertheless, the array splitting presents uswith the problem of determining the unique DOA for one source,which is more accurate among the others. To be precise, we list ourmain contributions as follows:

• A coprime EMVS array has been considered for DOAestimation.

• A minimum distance criterion is presented to uniquely estimatethe DOA based on PARAFAC decomposition of the receiveddata.

• Proofs of existence and uniqueness conditions of the minimumdistance are presented.

• A necessary and sufficient condition for a point to be theminimiser of the distance function is provided.

The organisation of the remainder of this paper is as follows.Section 2 works out a mathematical model of the received signalvia the coprime EMVS array. Section 3 formulates trilineardecomposition and the trilinear alternating least-square (TALS)updates of the received signal model. Section 4 presents themathematical formulation of the proposed algorithm for DOAestimation along with the proof of the fundamental theorem andpseudocode implementation. Section 5 provides simulation resultsto substantiate the effectiveness of the proposed method. Finally,Section 6 provides the related future work and the conclusion.

2 Data model2.1 Manifold of a single EMVS

Consider the presence of K uncorrelated monochromatic transverseelectromagnetic (TEM) waves, in a surveillance region which actsas a homogeneous isotropic medium and impinging on a single-isolated EMVS placed at the origin of the coordinate system asshown in Fig. 2.

Let the electric and magnetic field vectors of the kth completelypolarised electromagnetic wavefront from the angular direction (θk,φk) be ek and hk, respectively, where (r, θ, φ) denote the standardspherical coordinate system. In this situation, the manifold of asingle vector sensor is given by [8, 9]

μ =ek

hk=

cos φk cos θk −sin φk

sin φk cos θk cos φk

−sin θk 0−sin φk −cos φk cos θk

cos φk sin φk cos θk

0 sin θk

sin γkejηk

cos γk(1)

where 0 ≤ γk ≤ 2π is known as the auxiliary polarisation angle and−π ≤ ηk ≤ π is called polarisation phase difference [8, 9]. From (1),one can see that in contrast with spatially displaced arrays, thismanifold contains no spatial phase or time delay factors [8, 9].Thus, DOA estimates obtained via these manifolds areunambiguous.

2.2 Poynting vector for the kth source

For any electromagnetic source, its electric and magnetic fields arenot only orthogonal to each other but also to the source'snormalised Poynting vector which points along the direction ofpropagation. Thus, the components of the normalised Poyntingvector pk of the kth source are essentially the three directioncosines (uk, vk, wk) along the Cartesian coordinates [8, 9], i.e.

pk =px(θk, φk)py(θk, φk)

pz(θk)= ek × hk

∗ =uk

vk

wk

=sin θk cos φk

sin θk sin φk

cos θk

(2)

with ||pk|| = 1. So DOA of an electromagnetic source is uniquelydetermined by its normalised Poynting vector.

2.3 Signal model for coprime EMVS array

Consider a coprime EMVS array of M1 + M2 –1 (M1, M2 arecoprime integers) sensors placed along a line as shown in Fig. 1.Since a single EMVS outputs a six-dimensional (6D) vector, the6(M1 + M2 − 1) × 1 steering vector due to the kth source for thecoprime EMVS array is given as the following Kronecker product:

a(θk, φk, γk, ηk) = q(θk, φk) ⊗ μ(θk, φk, γk, ηk) (3)

where q(θk, φk) is (M1 + M2 − 1) × 1 vector containing inter-sensorspatial delays of the kth source. If K uncorrelated monochromaticTEM waves are illuminating this array, then the 6(M1 + M2 − 1) × Ksteering matrix A for the coprime EMVS array is given by thefollowing Khatri–Rao product [19]:

A = Q(θ, φ) ∘ U(θ, φ, γ, η) (4)

where ‘○’ denotes the Khatri–Rao product and

Q(θ, φ) = [q1 q2 … qK]U(θ, φ, γ, η) = [μ1 μ2 … μK]

(5)

In this scenario, the 6(M1 + M2 − 1) × 1 output for the coprimeEMVS array at time instant t is given by

x(t) = As(t) + n(t) (6)

Fig. 1 CPA as a superposition of two ULAs

Fig. 2 EMVS at the origin of coordinate systems

66 IET Radar Sonar Navig., 2019, Vol. 13 Iss. 1, pp. 65-73© The Institution of Engineering and Technology 2018

where s(t) ∈ ℂK × 1 is the source vector at time instant t andn(t) ∈ ℂ6(M1 + M2 − 1) × 1 is additive noise vector at sensor elements. Ifwe take L snapshots of the incoming signals, then we can write theabove equation as follows:

X = AS + N (7)

using (4) we can write

X = [Q ∘ U]S + N (8)

where X ∈ ℂ6(M1 + M2 − 1) × L is the received data matrix, S ∈ ℂK × L isthe source snapshot matrix and N ∈ ℂ6(M1 + M2 − 1) × L is the additivenoise matrix. Equations (6)–(8) give a general data model for acoprime EMVS array.

2.4 Uniform processing of signal model

The matrix A in (4) is a general steering matrix obtained via (5)and does not have any special structure. This is because the Qmatrix is formed with non-uniform spacing between the adjacentsensors of CPA, thus offering a great difficulty in analyses withconventional approaches. To deal with this difficulty, we split theCPA into two uniform linear sub-arrays such that the first ULA(ULA-1) contains M1 sensors with inter-sensor spacing of M2λ/2and the second ULA (ULA-2) contains M2 sensors with inter-sensor spacing of M1λ/2. This is shown in Fig. 1 for the case whenM1 = 5 and M2 = 7. Now, the received data model for thedecomposed CPA can be written in the form of two ULAs asfollows:

X = [Q ∘ U]S + N (9)

X′ = [Q′ ∘ U′]S + N (10)

where X, Q, U and N correspond to ULA-1 and X′, Q′, U′ and N′correspond to ULA-2. The matrices Q and Q′ now have aVandermonde structure and hence the models in (9) and (10) can beanalysed via conventional methods. Note that the source matrix isthe same for both the arrays.

3 Trilinear decomposition of received signalIn this section, we shall provide a trilinear decomposition of thereceived data. For ease of analysis, we shall only consider theULA-1 because results can be easily extended to ULA-2. The datamodel in (9) can be written in an alternative form and withoutnoise as follows [17, 19]:

X^ =

X^. .1

X^. .2

⋮X^

. . M1

= [Q ∘ U]S =

UD1(Q)UD2(Q)

⋮UDM1(Q)

S (11)

where Dm extracts the mth row of the argument matrix and returnsa diagonal matrix of that row. X^

. . m is the mth slice along the spatialdirection and is given by

X^. . m = UDm(Q)S (12)

The above equation can also be expressed as [20, 22]

x^m, c, l = ∑k = 1

Kqm, kμc, ksl, k (13)

where m = 1, 2 … M1, c = 1, 2 … 6, l = 1, 2 … L, qm,k are the (m,k)th entry of the direction matrix Q, μc,k is the (c, k)th entry of thepolarisation matrix U and sl,k is the (l, k)th entry of the signals

snapshots matrix S. The above form of the signal is commonlyknown as a trilinear model, trilinear decomposition or PARAFACanalysis model [18]. Equation (12) can be interpreted as slicing thethree-way array into a series of two-way arrays (also called slices)along the spatial direction. The symmetry of the trilinear model in(12) allows us to formulate two more systematic matrixrearrangements given by [17, 19]

Y^. . c = STDc(U)QT, c = 1, 2, …, 6 (14)

Z^. . l = QDl(ST)UT, l = 1, 2, …, L (15)

These arrangements can be understood as slicing the three-waydata along different directions. In particular, Y^

. . c is the cth slicealong the polarisation or component direction and Z^

. . l is the lthslice along the temporal direction.

3.1 Trilinear alternating LS

For the trilinear model in (12), we adopt LS approach to estimatethe matrices U, Q and S from noisy data X, i.e.

min ∥ X − X^ ∥F2 = min

Q, U, S∥ X − (Q ∘ U)S ∥F

2

where ||.||F denotes the Frobenius norm. By initialising the matricesQ and U to some fixed values, the above trilinear model islinearised in S and hence we can get an LS update for the sourcematrix S as follows [17]:

S~ = Q

~ ∘ U~ †

X (16)

where [.] denotes the generalised matrix inverse. The LS updatesfor the other two matrices can be obtained as follows [17, 19]:

Q~T = U

~ ∘ S~T †

Y (17)

U~T = S

~T ∘ Q~ †

Z (18)

According to (16)–(18), matrices S~, Q

~T and U~T are updated,

respectively, until a stopping criterion or a convergence conditionis satisfied [18]. This algorithm is commonly known as TALSs andthe updates are optimal when the additive noise is modelled asindependent and identically distributed (i.i.d.) Gaussian randomprocess [19]. A similar analysis can be applied to ULA-2 to get thefollowing TALS updates:

S~′ = Q

~′ ∘ U~′ †

X′

Q~′T = U

~′ ∘ S~′T †

Y′

U~′T = S

~′T ∘ Q~′

†Z′

(19)

In this paper, we have used the compressed factor algorithm [19,30] for fast implementation of TALS.

3.2 Model identifiability

A sufficient condition for the identifiability and unique PARAFACdecomposition is Kruskal's condition [20]. Consider (9) whichrepresents the unfolded matrix form of the three-way array X,Kruskal's condition states that the decomposition of X into threematrices Q, U and S is unique up to permutation and scalingambiguities, if but not necessarily [19, 20]

kQ + kU + kS ≥ 2(K + 1)


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where kQ, kU and kS give the k-rank [18] of matrices Q, U and S,respectively. To work out the identifiability of our model, we makethe following assumptions:

(A.1): Electromagnetic sources are zero mean and mutuallyuncorrelated stochastic process.(A.2): The additive noise is i.i.d. complex Gaussian process withvariance σ2, spatially white and non-polarised.(A.3): The sources have distinct DOAs.

Equation (5) along with the assumption (A.3) is enough toguarantee that Q is full column rank, i.e. kQ = K. Owing to (A.2)the signal matrix S also becomes a full rank matrix, so we have kS = K. Thus to achieve unique PARAFAC decomposition, Kruskalcondition requires

kU ≥ 2 (20)

In [20], it is proved that if (A.3) is satisfied, then (20) is alsosatisfied. This means that as long as the impinging sources (K inour case) have distinct DOAs, the PARAFAC decomposition of Xis unique.

4 Proposed algorithm for DOA estimationIn this section, we will discuss the proposed algorithm to estimateDOA. For simplicity, we will present the estimation of the sourceelevation only and assume that the azimuth angle is zero for allsources and polarisation parameters are known.

4.1 Minimum distance PARAFAC algorithm

The U~ update in (18) from ULA-1 can be written as

U~ = [u~1 u~2 … u~K] =

e~1x e~2x ⋯ e~Kx

e~1y e~2y ⋯ e~Ky

e~1z e~2z ⋯ e~Kz

h~

1x h~

2x ⋯ h~

Kx

h~

1y h~

2y ⋯ h~

Ky

h~

1z h~

2z ⋯ h~

Kz

(21)

From this matrix, we estimate the kth source Poynting vectorthrough vector cross-product of electric and magnetic fields asfollows:

p~k =p~x(θk, φk)p~y(θk, φk)

p~z(θk)= e~k × h

~k∗ =

e~kx

e~ky

e~kz

×h~

kx∗

h~

ky∗

h~

kz∗

(22)

Normalising the above Poynting vector, we obtain the directioncosines of the kth source from ULA-1

u~k

v~k

w~k

=sin θ

~k cos φ~k

sin θ~

k sin φ~k

cosθ~

k

= p~k∥ p~k ∥ (23)

Now, the elevation angle can be calculated as follows:

θkU = cos−1(w~k) (24)

The superscript ‘U’ indicates that the kth source elevation has beenestimated from U update [via (21)] of ULA-1. This same elevationcan also be estimated from the spatial phase factor update matrix Q

~

as follows [6]. The Q~ update in (17) from ULA-1 can be written as

Q~ =

q~0(θ1, φ1) q~0(θ2, φ2) ⋯ q~0(θK, φK)q~1(θ1, φ1) q~1(θ2, φ2) ⋯ q~1(θK, φK)

⋮ ⋮ ⋱ ⋮q~M1 − 1(θ1, φ1) q~M1 − 1

1 (θ2, φ2) ⋯ q~M1 − 1(θK, φK)

(25)

where q~m(θk, φk) gives the estimated spatial phase delay factorrelating the kth source to the mth EMVS of ULA-1 located at (xm,ym, zm). Normalising the kth column of above matrix by thecorresponding first entry and denoting the resulting column by z~k,we get

z~k = 1 q~1(θk, φk)q~0(θk, φk)

⋯ q~M1 − 1(θk, φk)q~0(θk, φk)

T(26)

Equation (26) gives the estimated spatial phase delay vector of thekth source corresponding to ULA-1. The spatial phase delay factorrelating the kth source to the mth EMVS of ULA-1 at the location(xm, ym, zm) is

qm(θk, φk) = e− j(2π /λ) xm ym zmukvkwk

= e− jΩk (27)

where m = 1, 2, …, M1 and Ωk = 2πd1 sin θk cos φk /λ. The actualspatial phase delay vector for the kth source corresponding toULA-1 is thus given by

q1(θk, φk) = 1 e− jΩk ⋯ e− j(M1 − 1)Ωk T (28)

Since the vectors in (26) and (28) refer to the same physicalquantity, they are essentially equal if the angles in two vectors aresame. Thus we have

gk = ∡(q1(θk, φk)) = 0 −Ωk ⋯ −(M1 − 1)Ωk (29)

and

g~k = ∡(z~k) = 0 α1 ⋯ αM1 − 1T (30)

Equating the two angle vectors, we get the following system ofequations:

Rω~ k = g~k (31)

where ω~ k = [0 Ω~ k]T and the matrix R is given by

R =

11⋮1

0−1⋮

−(M1 − 1)

(32)

An LS solution to the above system is given by

ω~ k =0

Ω~ k= (RTR)−1RTg~k (33)

where Ω~ k is the direction cosine of the source along the x-axis.Since the azimuthal angle is taken to be zero, therefore, the kthsource elevation can be obtained from the following relation:

θkQ = sin−1 λ

2πd1Ω~ k (34)

The superscript Q indicates that the kth source elevation has beenestimated from Q TALS update of ULA-1. Thus, we saw forULA-1 we have two choices to estimate DOA; either from thespatial phase factor matrix (34) or from the manifold of a singlevector sensor by estimating the Poynting vector (24). Similarly,


processing the ULA-2 and obtaining the PARAFAC updates of thematrices U′ and Q′, we can have two estimates of the kth sourceelevation from ULA-2 via the following relations:

θkU′ = cos−1(w~k′) (35)

and

θkQ′ = sin−1 λ

2πd2Ω~ ′k (36)

Thus, in this way we are available with a setA = {θk

U, θkQ, θk

U′, θkQ′} consisting of four estimates of the kth

source elevation. From these four estimates, we chose the onewhich is at the minimum distance from the range in which theactual DOA lies. For that purpose, we state and prove the followingtheorem.

Theorem 1: Let S ∈ ℝn be a non-empty closed convex set andθk be a point such that θk ∉ S & θk ∈ A. Then

a) There exists a unique point θ0 ∈ S which is at the minimumdistance from θk.

b) θ0 is the minimising point of S if and only if(θk − θ0)T(θ − θ0) ≤ 0, ∀θ ∈ S [31].

Definition 1: As S ⊆ ℝn is a convex set if for any, x1, x2 ∈ Sand any scalar α we have the following equation [31]:

αx1 + (1 − α)x2 ∈ S (37)

where 0 ≤ α ≤ 1 is the set of real numbers, a closed ball B − [x, r],a closed interval and an open ball B − (x, r) are the examples ofconvex sets [31].

Proof: (a) Existence: Let θk be any point which is not in the setS as shown in Fig. 3a. Let θ0 be any point inside S, then thedistance between θ0 and θk is given by

δ = infθ ∈ S

∥ θ0 − θk ∥ (38)

This distance is a function of θ. So we are minimising acontinuous function. By Weierstrass theorem [31], a continuousfunction over a closed bounded set always has extrema (minima ormaxima) that exists in the given set. In our case, S is given a closedset that is not necessarily bounded. To apply Weierstrass theoremto the above distance function, we construct a closed ball B [θk, 2δ]centred at θk and radius 2δ as shown in Fig. 3b. A closed ball is abounded set, and hence the intersection S ∩ B(θk, 2δ) is alsobounded set [31]. Since θ0 ∈ S ∩ B[θk, 2δ]; therefore, byWeierstrass theorem there exist a point θ0 which minimises theabove distance.□

Proof: (a) Uniqueness: To show the uniqueness of θ0 as theminimiser of the distance, consider another point θ′ ∈ S such that

∥ θ0 − θk ∥ = ∥ θ′ − θk ∥ = δ (39)

Since S is a convex set, then by definition the θ∗ = (θ0 + θ′)/2 ∈ Swith α = 1/2. Using triangular inequality we can write

2∥ θk − θ∗ ∥ = 2∥ θk − (θ0 + θ′)/2 ∥ ≤ ∥ θk − θ0 ∥ +∥ θk − θ′ ∥ = 2δ (40)

This implies that the distance between θ and θ* is less than orequal to δ. If this distance is strictly less than δ, then we reach at acontradiction because we have proved that θ0 is the minimiser.Thus, equality holds in the above expression indicating that θ0 andθ′ essentially represent the same point.□

Proof: (b) Let θ ∈ S and assume that (θk − θ0)T(θ − θ0) ≤ 0

∥ θk − θ ∥2 = ∥ θk − θ0 + θ − θ ∥2

⇒= ∥ θk − θ0 ∥2 + ∥ θ0 − θ ∥2 + 2(θk − θ0)T(θ0 − θ)(41)

Making use of the assumption that (θk − θ0)T (θ0 − θ) ≥ 0, we get

∥ θk − θ ∥2 ≥ ∥ θk − θ0 ∥2 (42)

which shows that θ0 is the minimising point. Conversely, supposeθ0 is the minimising point that is

∥ θk − θ0 ∥2 ≤ ∥ θk − θ̄ ∥2, ∀θ̄ ∈ S (43)

Since S is convex, then for any θ belonging to S we can write

αθ + (1 − α)θ0 ∈ S, ∀α ∈ [0, 1] (44)

Therefore

∥ θk − θ0 ∥2 ≤ ∥ θk − θ0 − α(θ − θ0) ∥2

⇒= ∥ θk − θ0 ∥2 + α2∥ θ − θ0 ∥2 − 2α(θk − θ0)T(θ − θ0)(45)

⇒ 2(θk − θ0)T(θ − θ0) ≤ α∥ θ − θ0 ∥2 (46)

Letting α → 0+, the result follows.□Rephrasing the results of the theorem in our context, it states

that if the range set or closed interval S = [θmin, θmax] to which thetrue DOA belongs is closed and convex, then for any point θkoutside this interval there always exists a unique point θ0 in Swhich is at a minimum distance from θk. Once this fact isestablished, the minimum distance (or the closest) DOA can beestimated using the condition given in part (b). To facilitate thealgorithmic implementations, we provide main steps of theproposed algorithm in Fig. 4.

4.2 Complexity analysis of the proposed algorithm

In this section, we analyse the computational complexity of theproposed algorithm by evaluating the number of floating pointoperations (flops) involved in computation of TALS updates of(16)–(18), vector cross-product direction finding (22), (23) and indirection finding via (33). We define a flop to be consisting ofeither a complex multiplication or a complex addition, regardlessof the fact that a complex multiplication involves four realmultiplications and two real additions, whereas a complex additionrequires only two real additions, making a multiplication moreexpensive than a summation. However, we count each operation asone flop. As earlier K, L and M represent the total number ofimpinging electromagnetic source waveforms, snapshots andvector sensors, respectively. Table 1 gives operation-wise flopsinvolved in evaluating a particular equation.

In PARAFAC decomposition, since matrix updates areiteratively computed until a convergence or stopping criterion is

Fig. 3 Existence of the distance minimiser(a) Set S and the point θk, (b) Intersection of set S and closed ball [θk, 2δ]


69

satisfied, these iterations usually are computationally intensive butoffer much better estimation performance as compared with otherexisting methods as will be cleared in the next section.

5 Numerical simulationsIn this section, we provide numerical examples to illustrate thesuperior performance of the proposed algorithm in terms of DOAestimation. For simulation setup, we consider a coprime EMVSarray with M1 = 5 and M2 = 7. The sources are modelled as randomGaussian processes and assume that each source has only one pathto EMVS array. The additive noise is modelled as a zero-meancomplex white Gaussian random process. For fair comparison, weconsider an equivalent uniform linear EMVS array with M1 + M2 − 1 = 11 vector sensors with spacing equal to half of thewavelength, to simulate the conventional PARAFAC, ESPRIT,long vector MUSIC (LV-MUSIC) [32, 33] and propagator method

[13] for vector sensor arrays. The range of the angle in thespectrum search in LV-MUSIC is within (0, 90) with a step size of0.001. The root-mean-square error (RMSE) in DOA estimation istaken as the performance metric and is defined as follows:

RMSE = 1KJ ∑

k = 1

K

∑j = 1

J(θ~k, j − θk)

2 (47)

where J is the total number of Monte Carlo trials, θ~

k, j representsthe estimated DOA of the kth signal source in the jth Monte Carlosimulation run. For each simulation scenario, 500 Monte Carlotrials are conducted.

Fig. 4 Main steps of the proposed algorithm

Table 1 Equation-wise flops for the proposed algorithmEquation Operation Flops Total complexity(16) Q ∘ U where Q ∈ ℂM × K and U ∈ ℂ6 × K 6MK K3 + (144M − 5)K2 + (72LM − 6L − 30M + 1)K per T ALS iteration

(Q ∘ U)T(Q ∘ U) where Q ∘ U ∈ ℂ6M × K 6K2(12M − 1)[(Q ∘ U)(Q ∘ U)]−1 K(K2 + K + 1)

[(Q ∘ U)(Q ∘ U)]−1(Q ∘ U) 36KM(2K − 1)[(Q ∘ U)(Q ∘ U)]−1(Q ∘ U)X, where X ∈ ℂ6M × L 6KL(12M − 1)

(17) U ∘ ST, where U ∈ ℂ6K and ST ∈ ℂL × K 6LK K3 + (144L − 5)K2 + (72LM − 6M − 30L + 1) K per T ALS iteration

(U ∘ ST)T(U ∘ ST), where U ∘ ST ∈ ℂ6L × K 6K2(12L − 1)[(U ∘ ST)(U ∘ ST)]−1 K(K2 + K + 1)

[(U ∘ ST)(U ∘ ST)]−1(U ∘ ST) 36KL(2K − 1)

[(U ∘ ST)(U ∘ ST)]−1(U ∘ ST), where Y ∘ ∈ ℂ6L × M 6KM(12L − 1)(18) ST ∘ Q, where ST ∘ ∈ ℂL × K and Q ∈ ℂM × K LMK K3 + (24LM − 5)K2 + (67LM − 35)K per T ALS iteration

(ST ∘ Q)T(ST ∘ Q), where ST ∘ Q ∈ ℂLM × K 6K2(2LM − 1)[(ST ∘ Q)(ST ∘ Q)]−1 K(K2 + K + 1)

[(ST ∘ Q)(ST ∘ Q)]−1(ST ∘ Q) 6KLM(2K − 1)

[(ST ∘ Q)(ST ∘ Q)]−1(ST ∘ Q)X, where X ∈ ℂLM × 6 36K(2LM − 1)(22) e~k

1 × (h~k1)∗, where e~k

1, h~

k1 ∈ ℂ3 12K 12K

(23) p~k1

∥ p~k1 ∥

, where p~k1 ∈ ℝ3

12K 12K

(33) R1TR1, where R1 ∈ ℝM × 2 48M − 24 98M − 28

(R1TR1)−1R1

T 36M + 8

(R1TR1)−1R1

Tg~k1, where g~k

1 ∈ ℝM 24M − 12


5.1 RMSE versus signal-to-noise ratio

We consider cases of one, two and three narrowband sources in thespace with DOAs at [10°], [10°, 15°] and [10°, 15°, 20°],respectively. The polarisation parameters of all the sources havebeen kept the same for this simulation. We record 500 snapshotsfor each source. Figs. 5–7 show the RMSE performance versusdifferent signal-to-noise ratios (SNRs) for the proposed and theother methods. It is clear that the proposed algorithm outperformsthe other algorithms including the simple average of four estimatesand the performance is consistent throughout the range of SNR. Atlow SNR, the estimates obtained via Q updates are better thanthose obtained via U updates and converse is true at high SNR.That is why the average estimation results in almost similarperformance at all SNRs. The reason for the slight degrading

performance of LV-MUSIC is that it breaks the local polarisedstructure of the data by stacking the outputs of component sensorsin the form of LVs, thus as opposed to its classical counterpartrequires a large number of snapshots for covariance matrixestimation at same SNR. Moreover, stacking the data in the form ofLVs also has the drawback of increased computations.

5.2 RMSE versus snapshots

We consider the case of one, two and three narrowband signalsources in the space with DOAs at [10°], [10°, 15°] and [10°, 15°,20°], respectively. The SNR is taken to be 0 dB. Again thepolarisation parameters of all the sources have been kept same.Unlike ESPRIT, LV-MUSIC and propagator method that requireestimation of the received signal covariance matrix, which dependson the collected number of snapshots, the proposed algorithmestimates the DOA by working directly on the received signalmeasurements. Figs. 8–10 show the RMSE performance versusdifferent number of snapshots for the proposed and the othermethods. Again the proposed algorithm performs superior to thesimple average of four estimates and the other algorithms.

5.3 Sources with different polarisations

In this section, we assess the performance of the proposedalgorithm for the case when sources have different polarisationparameters. We consider two sources: one having right circularpolarisation parameters with DOA of 10° and the other having leftcircular polarisation parameters with a DOA of 20°. Figs. 11 and12 show the RMSE performance versus different SNRs andnumber of snapshots at 10 dB SNR, respectively, for the proposedand the other methods. It can be seen that as opposed to theproposed, other methods work by breaking the polarisationstructure of the data resulting in degraded performance.

Fig. 5 RMSE versus SNR for one source

Fig. 6 RMSE versus SNR for two sources

Fig. 7 RMSE versus SNR for three sources

Fig. 8 RMSE versus snapshots for one source

Fig. 9 RMSE versus snapshots for two sources


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6 Conclusion and future workThis work presented a minimum distance-based DOA estimationalgorithm via PARAFAC analysis. We provided the mathematicalformulation of the problem and solved it using the trilineardecomposition of the received signal and obtained the DOAestimates via minimum distance criterion. Simulation resultsproved the efficacy of the proposed algorithm over the othermethods. To keep the mathematical formulation simple, wediscussed 1D DOA estimation only. 2D DOA estimation, jointangle and polarisation estimation via sparse vector SAP and therelated Cramer–Rao bound analysis are the topics of our futureresearch.

7 Acknowledgments

This work is supported by the China NSF Grant nos. 61371169,61601167 and 61601504, the Jiangsu NSF (BK20161489), theopen research fund of State Key Laboratory of Millimeter Waves,Southeast University (No. K201826), the Fundamental ResearchFunds for the Central Universities (No. NE2017103), GraduateInnovative Base (Laboratory) Open Funding of Nanjing Universityof Aeronautics and Astronautics (NUAA) (kfjj20170412), thePostgraduate Research and Practice Innovation Program of JiangsuProvince (KYCX18_0293) and the Funding of Jiangsu InnovationProgram for Graduate Education.

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