IET Signal Processing

Research Article

DOA estimation of multiple sources for amoving array in the presence of phase noise

ISSN 1751-9675Received on 4th February 2018Revised 12th May 2018Accepted on 16th July 2018E-First on 8th October 2018doi: 10.1049/iet-spr.2018.5143www.ietdl.org

Zhan Shi1 , Xiaofei Zhang1,2, Le Xu1

1College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, People's Republic of China2State Key Laboratory of Millimeter Waves, Southeast University, Nanjing, People's Republic of China

E-mail: sz__yz@163.com

Abstract: The authors investigate the issue of direction of arrival (DOA) estimation in the presence of phase noise for multiplesources with a synthetic linear array, which is synthesised by a short moving array. The extended towed array measurements(ETAM) method can extend the array aperture greatly, but can only be applied to multiple coherent sources and requires thearray to move with constant velocity. To tackle the problems, the authors generalise the ETAM method to multiple incoherentsources and array non-uniform motion case. The authors first formulate the manifold of the extended synthetic array (SA)moving with known velocity in a straight line. Then the initial DOA estimates and phase correction factors are estimatedsuccessively by two-dimensional multiple signal classification (2D-MUSIC) spectrum search using adjacent measurementssampled by the moving array. Moreover, to reduce the complexity, the authors also propose a reduced dimensional MUSIC (RD-MUSIC) method to turn the two-dimensional peak search to one-dimensional. As the array aperture is extended by propercompensation, the DOA estimation performance of the proposed SA method improves. Besides, the proposed method canresolve more number of sources than sensors since every two measurements are used for estimation in each process.Simulation results validate the effectiveness of the proposed method.

1 IntroductionDirection of arrival (DOA) estimation is one of the main issues inarray signal processing and has vital applications in radar, sonarand wireless communication [1–5]. Nowadays, to detect sources inthe complex environment [6, 7] with all kinds of interferences,high-resolution DOA estimation methods are highly required. Themost common way to improve estimate resolution is extending thearray aperture, which will in turn increase the burden of bearingplatforms unfortunately. In such a situation, the synthetic array(SA) methods are proper solutions to meet both the high-resolutionand miniature platform requirements [8, 9].

SA methods, which exploit platform motion and signaltemporal coherence to extend the array aperture, are widely used inelectromagnetic and acoustic signal processing, such as syntheticaperture radar (SAR) and synthetic aperture sonar (SAS) [10, 11].The basic idea of SA is to treat the signals received by one sensorat different moments as signals received by different sensors at thesame moment through signal processing. Thus large aperture arrayscan be obtained using relatively shorter arrays. The concept of SAwas first proposed by Williams [12] and then analysed theimplementation and performance in detail by Autrey [13]. Due tothe advantages of SAs in the aperture extension aspect, numerousresearches on SA have been proposed. Among these methods, theextended towed array measurements (ETAM) [14–16], the fastFourier transform synthetic aperture (FFTSA) method [17] and themaximum likelihood (ML) method [18] are paid the most attention.It was shown that the ETAM method takes advantages over theothers in estimation performance, array gain and some otheraspects [19]. The ETAM method compensates the signals receivedby moving towed array in successive measurements coherentlywith a phase correction factor, which is caused by array motion anduncertain phase noise. Recently, many improved ETAM methodshave been proposed [20, 21]. As the original ETAM method needsthe hydrophones to be overlapped between successivemeasurements which limits the array motion, an estimation methodwithout restriction of array position overlap was proposed in [20].Wang et al. [21] solved the problem of ‘deviation interval’ aboutthe ETAM method by taking the average of normalised cross-

correlation instead of cross-correlation phases. Besides, the conceptof ETAM is also applied for other arrays. Ramirez and Krolik [22]exploited the co-prime array motion to increase the degrees offreedom (DOFs) where the conventional co-prime array is not ahole-free linear array. The near-field sources localisation wasconsidered in [23] using a moving array together with receivedsignal strength information.

Throughout the studies mentioned earlier, the towed array isusually considered to move with constant velocity, which is hard tomaintain in practice. Besides, for the case of multiple sources, theoriginal ETAM method [16] requires all sources to be coherentover the whole synthetic period since only one phase correctionfactor can be estimated by two successive measurements [24].However, the uncertain phase noise [25] caused by systemic erroror other effects, such as the random initial phase, for differentsources may be different and it will degrade the estimationperformance if it is not correctly compensated [16].

In this paper, we present a DOA estimation method consideringphase noise for multiple incoherent sources with the large aperturelinear array synthesised by a moving array. First, the model of themoving linear array considering phase noise is formulated and themanifold of the extended SA is derived. Then, the phase correctionfactors and initial DOA estimates are estimated through two-dimensional multiple signal classification (2D-MUSIC) spectrumsearch using two adjacent measurements received by the movingarray. We can compensate the phase difference caused by arraymotion and uncertain phase noise and form an extended aperturearray once phase correction factors of all sources in the wholesynthetic process are estimated. Furthermore, we also present areduced-dimensional (RD) method [26] to reduce the complexityof the 2D spectrum search but with little performance loss.Cramer–Rao bound (CRB) of the SA is derived for comparison.The DOA estimation performance improves as the aperture islargely extended. According to the simulations, the proposedmethod with a 20-elements synthetic uniform linear array (ULA)formed by a 4-elements moving array has similar estimationperformance with conventional MUSIC algorithm [27] with a 14-elements physical ULA. The performance loss of the SA comparedto its corresponding physical array is because the proposed method

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takes the phase noise into consideration and thus has moreunknown parameters to estimate than the ideal MUSIC. Inaddition, the proposed method can resolve more number of sourcesthan sensors because two successive measurements are used foreach estimation process.

The main contributions of this paper can be summarised asfollows. (i) We formulate the SA model where the linear arraymoves with varying velocity and the phase noise of multiplesources is considered, which is more general in practice. (ii) Wepresent a joint successive phase correction factors and DOAestimation algorithm. Once the phase correction factors areestimated, we can form an extended aperture array, with which wecan improve the estimation performance and achieve more DOFs.(iii) To reduce the computational complexity, we present an RD-MUSIC method, which has the similar performance to the two-dimensional one.

The remainder of this paper is organised as follows. Section 2introduces the signal model of the SA. In Section 3, we derive theproposed 2D and RD DOA estimation algorithm with the syntheticlinear array, and analysis of the proposed method is given inSection 4. Section 5 shows several numerical simulations and weconclude this paper in Section 6.

Notations: Throughout this paper, lower-case and upper-casebold characters stand for vector and matrix. ( ⋅ )T, ( ⋅ )∗ and ( ⋅ )H

represent the transpose, conjugate and conjugate transposeoperation of a vector or matrix. ⊙ and ⊗ stand for Khatri–Raoproduct and Kronecker product. trace{ ⋅ } represents the trace of amatrix.

2 Data modelConsider an M-elements linear array moving along the x-axis withany known velocity v, as shown in Fig. 1. Assume there are K far-field uncorrelated narrowband sources impinging on the array frombearings θk ∈ ( − π /2, π /2) with frequencies f k, k = 1, …, K. Thepositions of the elements of the linear array are defined asdm, m = 1, 2, …, M and the minimum inter-element spacingΔdmin < λmin/2, where λmin is the smallest wavelength among the Ksources and d1 = 0 is set as the reference. The received signal at themth sensor of the stationary array can be denoted as

xm(t) = ∑k = 1

Ksk(t)e j2π f kτm(θk) + wm(t) (1)

where sk(t) = αk(t)e j(2π f kt + θk(t)) is the kth narrowband signal, αk(t)and θk(t) are the slowly changing amplitude and phase modulationfunction, τm(θk) = dmsin θk /c is the delay relative to the reference,wm(t) is the additive white noise with zero mean and variance σ2.Under the narrowband assumption, αk(t) ≃ αk(t − τm) andθk(t) ≃ θk(t − τm) hold, then the kth received signal at the mthsensor sk(t − τm) can be approximately as sk(t)e j2π f kτm. It is assumedthat signals meet the narrowband property during the integrationperiod.

Define unit measurement period as T and sample successive Lmeasurements of data, for example, in Fig. 1, the first measurementbegins at T1. During each measurement, sample the array N timeswith the sampling interval τ(Nτ < T). Then, at a time instant t + Tl,the data received by the mth element in the lth measurement can berepresented as [14, 17] (see (2)) where t ∈ [0, T], Tl = (l − 1)Tstands for the start time of lth measurement, f k′ = f k(1 + vlsin θk /c)is the frequency of the kth source along with Doppler shift, c is thepropagation velocity of signals, φki represents the uncertain phasenoise caused by systemic errors or other effects of the kth source inthe ith measurement and set φk1 = 0 as the reference, sk(t) standsfor the signal received by the reference element in the firstmeasurement, wm(t + Tl) is the additive white noise with zero meanand variance σ2. Note that, here we do not require the velocity to beconstant, it means the array can move with varying velocity in astraight line. For simplicity, assume the velocity during eachmeasurement does not change a lot, i.e. the velocity vl in the lthmeasurement is taken as constant.

The following assumptions are made throughout the entirepaper. (i) The number of sources K and sources frequencies areassumed to be known. If not, the existing information theoreticcriteria [28] or MDL method [29] can be adopted to estimate K andFFT techniques [30, 31] to estimate frequencies in advance. (ii)The bearings do not change during the whole integration periodunder the far-field sources assumption. (iii) The synthetic durationis less than maximum signal temporal coherence, which will befurther analysed in Section 4. (iv) The array is rigid and thus allsensors move in the same direction with no vibration errors.

Usually, the wave propagates much faster than the movingarray, i.e. vl ≪ c, then vldmsin2 θk /c2 ≪ dmsin θk /c holds and theterm vldmsin2 θk /c2 can be omitted. As a result, (2) can be simplifiedas (see (3)) where Dl = vlTl is the total distance the array moves

Fig. 1  Synthetic moving linear array model

xm(t + Tl) = ∑k = 1

Ke j2π f ′k(Tl + (dm/c)sinθk) + j∑i = 1

l φkisk(t) + wm(t + Tl)

= ∑k = 1

Ke j2π f k(1 + (vlsinθk /c))(Tl + (dm/c)sinθk) + j∑i = 1

l φkisk(t) + wm(t + Tl)(2)

xm(t + Tl) = ∑k = 1

Ke j2π f k(Tl + ((dm + Dl)/c)sinθk) + j∑i = 1

l φkisk(t) + wm(t + Tl) (3)

30 IET Signal Process., 2019, Vol. 13 Iss. 1, pp. 29-35© The Institution of Engineering and Technology 2018

from the beginning to the lth measurement if the velocity v isconstant, otherwise Dl = ∑i = 1, …, l viT and D1 = 0.

3 Proposed DOA estimation method with asynthetic linear array3.1 Array aperture extension

Assume there exists an element of the physical array, which islocated at dm + Dl, written as the m′th element, then the signalreceived at instant t ∈ 0, τ by this element can be represented as

xm′(t) = ∑k = 1

Ke j2π f k((dm + Dl)/c)sin θksk(t) + wm′(t) . (4)

Without considering the additive white noise, the signal receivedby the mth element of the moving array xm(t + Tl) and the signalreceived by the m′th element of the physical array xm′(t) have thefollowing relationship:

xm′(t) = ∑k = 1

Ke− j2π f kTl − j∑i = 1

l φkixm(t + Tl)

= ∑k = 1

Ke jψklxm(t + Tl)

(5)

where ψkl = − 2π f kTl − ∑i = 1, …, l φkl is the phase difference causedby measurement delay and phase noise.

From (5), we can conclude that after compensating for thephase difference, the signals received by the moving array can beseen as received by the equivalent element of the physical array,which means the number of array elements and array aperture canbe enlarged by this way.

Then, the signals of all L measurements received by the movingarray can be represented in matrix form as

X = ASAS + W (6)

where ASA = asa(θ1), …, asa(θK) ∈ ℂML × K is the manifold of thesynthetic linear array, (see equation below) l = 1, …, L is thesteering vector of the moving array in the lth measurement,S = [s1, …, sK] ∈ ℂK × N is the signal matrix, sk = [sk(0), …,sk(N − 1)]T ∈ ℂN × 1, W is the noise matrix.

Using L measurements, the original M-elements physical lineararray can be extended to an ML-elements linear array withM(L − 1) virtual elements at most if no virtual element overlaps,thus the DOFs and estimation performance will be largelyincreased. However, processing all these measurements directlyrequires the signal to be coherent over the whole integration period,which is hard to maintain [16]. Thus we estimate the parameterssuccessively using two adjacent measurements, which will beshown in Section 3.2.

3.2 Phase correction factor and initial DOA estimation with2D-MUSIC

It is clear from (5) that the key process to extend the array apertureis the compensation for the fluctuation of the phase difference.There are two ways to estimate the phase differences, estimating allthese phase differences from the L measurements or estimating partof them from two adjacent measurements successively. The formerway needs each signal to be coherent over the whole process whichis discussed in Section 4.1 and is hard to meet, while the latter oneonly requires the signal to be coherent between two adjacentmeasurements. Thus we apply the latter method, i.e. estimating thephase differences successively.

According to (3), the signals received by the mth element in thelth measurement and l + 1th measurement of the kth source havethe following relationship:

xm(t + Tl + 1) = e j2π f kvl + 1Tsinθk /ce j(2π f kT + φk, l + 1)xm(t + Tl) (7)

And the received data in the lth measurement can be written inmatrix form as

Xl = ASl + Wl (8)

where A = a(θ1), …, a(θK) ∈ ℂM × K is the manifold of thephysical array,a(θk) = [1, e j2π f kd2sin θk /c, …, e j2π f kdMsin θk /c]T ∈ ℂM × 1, k = 1, …, K,Sl ∈ ℂK × N and El ∈ ℂK × N are the signal and noise matricesreceived by the reference element in the lth measurement.

From (7) and (8), without considering the additive noise we canget

Xl + 1 = ASl + 1 = a(θ1)p1l, …, a(θK)pKl Sl (9)

where pkl = e j2π f kvl + 1Tsin θk /ce j(2π f kT + φk, l + 1) = e jϕkl, k = 1, …, K andϕkl is called the ‘phase correction factor’, similar to that in [16]. Itis obvious that the phase correction factor ϕkl involves arraymotion and phase noise.

Then, consider (8) and (9) jointly and we can obtain

X′l =Xl

Xl + 1=

ASl

ASl + 1= Pl ⊙ A Sl = BlSl (10)

where Pl = p1l(ϕ), …, pKl(ϕ) ∈ ℂ2 × K,

pkl(ϕ) = 1, pklT = 1, e jϕkl T

, ⊙ stands for Khatri–Rao product,Bl = Pl ⊙ A can be seen as the equivalent manifold of the movingliner array using two measurements of data during the period fromTl to Tl + 1.

During this period, the data covariance matrix can berepresented as [1, 2]

Rl = X′lX′lH/N = UslΛslUsl

H + UnlΛnlUnlH (11)

where Usl and Unl stand for the signal subspace and noise subspace,respectively, Λsl is a diagonal matrix consisting of K biggesteigenvalues λ1, …, λK and Λnl consists of the rest 2M − Keigenvalues.

The phase correction factors and initial DOA estimates can beobtained according to the following function [26]:

min(ϕ, θ)

V(ϕ, θ) = min(ϕ, θ)

bkH(ϕ, θ)UnlUnl

Hbk(ϕ, θ) (12)

Then, the 2D-MUSIC spatial spectrum can be described as [27]

Pl(ϕ, θ) = 1bk

H(ϕ, θ)UnlUnlHbk(ϕ, θ) (13)

where bk(ϕ, θ) is the kth column vector of the manifold Bl, ϕ and θare the grid of the phase correction factor and initial DOAestimates. We can obtain K phase correction factors [ϕ^

1l, …, ϕ^Kl]

and K initial DOA estimates by locating K largest peaks of the 2Dspectrum.

Notably, here we only use two measurements, hence the DOAestimation may not be precise. Once all L − 1 pairs phasecorrection factors are estimated, we can get L corrected

asa(θk) = a1, saT (θk), …, aL, sa

T (θk) T, al, sa(θk) = e j2π f k(d1 + Dl)sin θk /ce jψkl, …, e j2π f k(dM + Dl)sin θk /ce jψkl T ∈ ℂM × 1,

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measurements and the accurate DOA estimation can be obtainedfrom the whole receiving signal.

3.3 Reduced dimensional method

To reduce computational complexity, we can utilise the one-dimensional spectrum search of the RD-MUSIC instead of theexpensive 2D-MUSIC search.

The function V(ϕ, θ) in (12) can be rewritten as [26]

V(ϕ, θ) = pkl(ϕ) ⊗ a(θ) HUnlUnlH pkl(ϕ) ⊗ a(θ)

= aH(θ) pkl(ϕ) ⊗ I2HUnlUnl

H pkl(ϕ) ⊗ I2 a(θ)= aH(θ)Q(ϕ)a(θ)

(14)

where Q(ϕ) = pkl(ϕ) ⊗ I2HUnlUnl

H pkl(ϕ) ⊗ I2 , I2 is a 2-dimensional identity matrix.

According to [26], (12) can be represented as (15) which is aquadratic optimisation problem

minϕ

aH(θ)Q(ϕ)a(θ), s . t . e1Ha(θ) = 1 (15)

where the constraint is to eliminate the trivial solution a(θ) = 0,e1 = 1, 0, …, 0 T ∈ ℝM × 1 and the steering vector a(θ) is obtainedby [26]

a(θ) = Q−1(ϕ)e1

e1HQ−1(ϕ)e1

(16)

Then, the phase correction factor ϕ can be estimated via

ϕ^ = arg max e1

HQ−1(ϕ)e1 (17)

With the estimated ϕ^ and (16), the steering vector a^(θ) can be

obtained and we can utilise Least Squares (LS) method to estimateDOA by

minck

∥ Hck − gk ∥F2

(18)

where gk = − angle(a^(θk)), ck = ck0, ck1T and ck0 is an arbitrary

constant, ck1 = sin θk, H = 1M, dk ∈ ℝM × 2, 1M ∈ ℝM × 1 is acolumn vector whose elements are all 1,dk = 2π f kd1/c, …, 2π f kdM /c T.

For one estimated ϕ^k, its corresponding θ

^k can be obtained

through (18), thus the DOA estimate and phase correction factorare auto-paired for each joint estimation process.

Though the initial DOA estimated by θ^k = sin−1(c^k1) may not be

very precise, it can still be used to pair the phase correction factorsof different sources.

3.4 Fine DOA estimation

Till now, L − 1 pairs phase correction factors [ϕ^2, …, ϕ

^L],

ϕ^l = [ϕ^

1l, …, ϕ^Kl]T, l = 1, 2, …, L of K sources are obtained from

L measurements. Then these factors can be used to compensate themanifold of the SA and the spatial spectrum function can berepresented as

Psa(θ) = 1a^sa

H (θ)U^nU

^nHa^sa(θ) (19)

where a^sa(θk) = a^1, saT (θk), …, a^L, sa

T (θk)T, k = 1, …, K and

a^l, sa(θk) = e j2π f kd1sin θk /c∑i = 1l p^ ki, …, e j2π f kdMsin θk /c∑i = 1

l p^ kiT,

p^ ki = e jϕ^ki, U^n is the noise subspace of the covariance matrix R^

x ofthe received data X in (6), R^

x = XXH/N.

3.5 Detailed steps

We have obtained the DOA estimation with the synthetic lineararray via the proposed method and the detailed steps are providedas follows.

Step 1: Estimate L − 1 pairs phase correction factors [ϕ^2, …, ϕ

^L]

using two adjacent measurements through the 2D-MUSIC or RD-MUSIC methodsStep 2: Compute the covariance matrix R^

x of the whole receivedsignals X in (6) and get the noise subspace U^

n

Step 3: Obtain the fine DOA estimation by 1D spatial spectrumsearch by (19).

4 Performance analysis4.1 Temporal coherence

It is assumed in Section 2 that the integration period of twosuccessive measurements is less than maximum signal temporalcoherence, which means the signals received in two measurementsare approximately coherent so that we can process them jointly.Generally, the signal temporal coherence usually degrades becauseof the change of the source frequency or phase noise [16, 32]. Asthe phase noise will be estimated in the estimation process, here weonly consider the influence of the frequency change resulting fromthe Doppler spread. The coherence period Tc is inverselyproportional to Doppler spread [32]

Tc = 1f d

= 1f 0vsin θ /c = λ0

vsin θ (20)

where f 0 and v are the signal frequency and array velocity.It can be seen that the coherence period and the time needed to

synthesise one virtual element λ/2v have the same level.According to the simulations in [15, 20], even when the integrationperiod is several times of the coherence period, the SA method stillworks. Thus the assumption of signal coherence over the syntheticperiod can be satisfied approximately.

4.2 Complexity, aperture extension and DOF

The complexity of the successive 2D-MUSIC spectrum search andcorresponding RD-MUSIC is analysed, where we only consider thecomplex multiplication. For 2D-MUSIC, the complexity isO 4M2N + 8M3 + n2 ⋅ 4M(2M − K) , and for RD-MUSIC isO 4M2N + 8M3 + n ⋅ (2M + 1)(2M − K) , where n is the times ofone-dimensional spectrum search. It is clear that the RD-MUSICmethod has a much lower complexity as it only needs a one-dimensional search.

According to the above analysis, we can see that after synthesis,the number of elements of the SA is extended to ML at most, andthe maximum array aperture is ∑l = 1

L vlTl + D, where D is theoriginal aperture of the physical linear array. For simplicity, herewe ignore the situation that two synthetic elements overlap in thesame position. In practice, the number of unique elements dependson motion velocity and unit measurement period T.

In the process of phase correction factor estimation, we use thedata of two measurements and the dimension of the manifold Bl is2M × K, so the maximum number of the sources that can beresolved for 2D-MUSIC is 2(M−1), which is more than the numberof sensors. For RD-MUSIC, as not all information is used inV(ϕ, θ) when estimating the phase correction factor, the DOF maybe less than 2D-MUSIC, but it can still resolve at least M sources.

4.3 Cramer–Rao bound

The CRB is the lower bound of unbiased parameter estimation. Inthis subsection, we derive the CRB of the proposed method withthe synthetic linear array. For simplicity, assume that the noisevariance σ2 = 1 and the signal covariance matrix Rs = SSH/N are

32 IET Signal Process., 2019, Vol. 13 Iss. 1, pp. 29-35© The Institution of Engineering and Technology 2018

known, then there are LK parameters in total to estimate (i.e. KDOAs and (L − 1)K phase correction factors), written as

ξ = θ1, …, θK, ϕ21, …, ϕ2K, …, ϕLKT (21)

The CRB of DOA and phase correction factor can be representedas [33]

E (ζ^ − ζ0)(ζ^ − ζ0)

T ≥ CRB (22)

CRB = F−1 (23)

where ζ^ is the estimate parameters vector, F ∈ ℂLK × LK is the

Fisher information matrix, which can be blocked as

F =Fθθ Fθϕ

Fϕθ Fϕϕ(24)

where Fθθ and Fϕϕ stand for the DOA estimation block and phasecorrection factor estimation block, respectively. According to [33],the (m,n)th element of F can be represented as

Fmn = N ⋅ trace R−1 ∂R∂ζm

R−1 ∂R∂ζn

= 2N ⋅ Re trace(DmRsAsaH R−1AsaRsDn

HR−1)

+trace(DmRsAsaH R−1DnRsAsaR−1)

(25)

where R = XXH/N, Dm = ∂Asa/∂ζm, Dn = ∂Asa/∂ζn.

4.4 Advantages

The advantages of the proposed DOA estimation method with thesynthetic linear array are summarised as follows.

i. The proposed method can be applied to arbitrary linear arrayregardless of array motion.

ii. The array aperture is largely extended by array motion, thusthe DOA estimation performance is improved compared to theoriginal physical array.

iii. The proposed RD-MUSIC method can reduce complexity andachieve similar performance with 2D-MUSIC spectrum searchmethod.

5 SimulationsIn this section, we present numerical simulations to verify theeffectiveness of the proposed DOA estimation method with thesynthetic linear array. For the ease of performance comparisonbetween the synthetic array and physical array, we consider thesynthetic moving ULA case and set vlT = Mλ/2, e.g. during eachmeasurement the distance the array moves equals M times half-wavelength. Assume the phase noise is uniformly distributedbetween −π and π.

Define the root mean square error (RMSE) as the performancecomparison metric:

RMSE = 1CK ∑

c = 1

C

∑k = 1

K(θ^c, k − θk)

2(26)

where C stands for the times of Monte Carlo simulations and θ^c, k is

the estimation of the kth bearing for the cth trial. In this paper, weset C = 500.

5.1 Phase correction factor estimation and validation ofincreased DOFs

In this simulation, we illustrate the process of phase correctionfactor estimation and show that the proposed method can resolve

more sources than sensors. Assume four sources from the bearings[10∘, 20∘, 30∘, 40∘] impinging on a 3-elements ULA with half-wavelength inter-element spacing and set N = 200, SNR = 20 dB.Fig. 2 depicts the contour figure of the 2D-MUSIC spectrum ofinitial DOA and phase correction factor using the first and secondmeasurements according to (12) in Section 3.2. It is clear that all 4sources can be detected as the 3-elements moving array can resolve2(M − 1) = 4 sources at most according to Section 4.2. By two-dimensional peak search, we can get four initial DOA estimatesand four phase correction factors corresponding to each source.Through L measurements, the above process needs to be done L − 1times and then the manifold of the SA Asa can be compensated foraccurate DOA estimation.

Fig. 3 shows the phase correction factors estimationperformance versus SNR, where θ = [10∘, 20∘], M = 4, L = 5 and thephase noise is uniformly distributed between −π and π. The RMSEof phase correction factors is defined as

RMSE = 1CK(L − 1) ∑

c = 1

C

∑k = 1

K

∑l = 2

Lϕ

^c, k, l − ϕk, l

2(27)

where ϕ^c, k, l is the estimate of the kth source in the lth measurement

in the cth trial and ϕk, l is its corresponding real value. As we setϕk, 1 = 0 as a reference, we only need to estimate (L − 1) pairs ofphase correction factors.

With the increase of SNR, the estimation performance of phasecorrection factors improves, which will achieve more accuratephase difference compensation and improve the DOA estimation aswell.

5.2 Comparison of DOA estimation RMSE versus SNR

We compare the estimation performance of the proposed methodwith the synthetic ULA and conventional MUSIC algorithm [27]with the physical ULA. In this simulation, for a fair comparison,

Fig. 2  Phase correction factor and initial DOA estimation

Fig. 3  Phase correction factors estimation performance versus SNR

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assume all ULAs are with half-wavelength inter-element spacingd = λ/2. The 4-elements moving ULA is measured five times, e.g.M = 4, L = 5. With the assumption vlT = Mλ/2, the moving arraysynthesises a 20-elements ULA. Assume N = 200 andθ = [10∘, 20∘]. The CRB of synthetic and physical arrays is derivedin Section 4.4 and [33], respectively.

We can conclude from Fig. 4 that the CRB and estimationperformance of the 2D and RD methods (represented as CRB SA,Proposed RD and Proposed 2D) with the 20-elements syntheticULA is similar to ideal MUSIC with the 14-elements physicalULA (represented as CRB M = 14 and ULA M = 14) and worsethan that of a 20-elements physical ULA (represented as CRB M = 20 and ULA M = 20). The performance loss compared to a 20-elements physical ULA is because the proposed method takes thephase noise into consideration and has to estimate the phasecorrection factor, thus the number of unknown parameters (LK) ismuch more than conventional MUSIC algorithm with physicalULA (K). However, it is still remarkable that a 4-elements movingarray can have comparable performance with a 14-elementsphysical array by the synthetic method, which validates theeffectiveness of the SA method. Besides, we can also notice thatthe RD method has the similar performance with the 2D method,but with much lower complexity.

5.3 Comparison of DOA estimation versus snapshot (N)

Fig. 5 shows the DOA estimation performance of the proposedmethods with the synthetic ULA and MUSIC algorithm with thephysical ULA, where the 20-elements synthetic ULA issynthesised with a 4-elements moving ULA through 5measurements, SNR = 10 dB and θ = [10∘, 20∘]. Here, we have thesame conclusions as Section 5.2 that the proposed method with 20-elements synthetic ULA has similar performance to MUSIC withthe 14-elements physical ULA and the proposed RD method hassimilar performance with the 2D method. All methods achievebetter estimation with the snapshots increasing.

5.4 Comparison of DOA estimation versus the number ofarray elements (M)

Fig. 6 shows the DOA estimation performance comparison versusthe number of moving array elements, where the synthetic ULA issynthesised through five measurements, θ = [10∘, 20∘] and N = 200.It is evident that the estimation performance improves with theincrease of the number of array elements and SNR as moreelements mean higher diversity gain.

5.5 Comparison of DOA estimation versus the number ofmeasurements (L)

Fig. 7 shows the DOA estimation performance comparison versusthe number of measurements, where the 4-elements moving ULAis used to synthesise, N = 200 and θ = [10∘, 20∘]. It shows clearlythat the performance of the algorithm improves with L increasing,because large measurement number can bring more equivalent

array elements and resultantly improve the estimation accuracy.However, more measurements of the moving array will cost longerintegration periods, so the number of measurements needs to bechosen reasonably.

6 ConclusionIn this paper, we propose a DOA estimation algorithm consideringphase noise for multiple sources with the synthetic linear array,which is synthesised by a short moving array. The proposedalgorithm does not require the integration array to move in constantvelocity or require multiple sources to be coherent over theintegration period. Hence, it can be seen as a generalisation of theoriginal ETAM method. In the proposed algorithm, the phasecorrection factors caused by phase noise and array motion areestimated through two adjacent measurements received by themoving array and with which the manifold of a large aperture SAcan be formed after properly compensating. As the array aperture isextended, the DOA estimation performance is also improved.

Fig. 4  Comparison of DOA estimation performance versus SNR

Fig. 5  Comparison of DOA estimation performance versus N

Fig. 6  Comparison of DOA estimation performance versus M

Fig. 7  Comparison of DOA estimation performance versus L

34 IET Signal Process., 2019, Vol. 13 Iss. 1, pp. 29-35© The Institution of Engineering and Technology 2018

Besides, the proposed algorithm can resolve more sources thansensors. According to the simulations, the proposed method with a20-elements SA formed by a 4-elements moving array can havesimilar performance with the ideal MUSIC algorithm with a 14-elements physical array, the reason the performance of SA is worsethan its corresponding physical array is that during the synthesisprocess, the SA method needs to estimate more unknownparameters. Furthermore, we also propose an RD method to avoidthe expensive complexity of the two-dimensional peak search buthas little performance loss.

7 AcknowledgmentsThis work is supported by China NSF Grants (61371169,61601167, 61601504), Jiangsu NSF (BK20161489) and GraduateInnovative Base (laboratory) Open Funding of Nanjing Universityof Aeronautics and Astronautics (kfjj20170412).

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