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Telecommun Syst (2015) 59:93–110DOI 10.1007/s11235-014-9886-3

An energy efficient DOA estimation algorithm for uncorrelatedand coherent signals in virtual MIMO systems

Liangtian Wan · Guangjie Han · Joel J. P. C. Rodrigues ·Weijian Si · Naixing Feng

Published online: 19 November 2014© Springer Science+Business Media New York 2014

Abstract The multiple input and multiple output (MIMO)and smart antenna (SA) technique have been widely acceptedas promising schemes to improve the spectrum efficiencyand coverage of mobile communication systems. The defi-nition of direction-of-arrival (DOA) estimation is that mul-tiple directions of incident signals can be estimated simul-taneously by some algorithms using the received data. Theconventional DOA estimation of user equipments (UEs) isone by one, which is named as sequential scheme. The Vir-tual MIMO (VMIMO) scheme is that the base station (BS)estimates the DOAs of UEs in a parallel way, which adoptsthe SA simultaneously. Obviously, when the power is fixed,the VMIMO scheme is much more energy efficient than thesequential scheme. In VMIMO scheme, a set of UEs are

L. Wan · W. Si (B)Department of Information and Communication Engineering,Harbin Engineering University, Harbin 150001, Chinae-mail: [email protected]

L. Wane-mail: [email protected]

G. Han (B)Department of Information and Communication Systems,Hohai University, Changzhou 213022, Chinae-mail: [email protected]

J. J. P. C. RodriguesInstituto de Telecomunicações, University of Beira Interior,6201-001 Covilhã, Portugal

J. J. P. C. RodriguesUniversity of ITMO, St. Petersburg 197101, Russiae-mail: [email protected]

N. FengInstitute of Electromagnetics and Acoustics, Xiamen University,Xiamen 361005, Chinae-mail: [email protected]

grouped together to simultaneously communicate with theBS on a given resource block. Then the BS using multi-ple antennas can estimate the 2D-DOA of the UEs in thegroup simultaneously. Based on VMIMO system, the 2D-DOA estimation algorithm for uncorrelated and coherent sig-nals is proposed in this paper. The special structure of mutualcoupling matrix (MCM) of uniform linear array (ULA) isapplied to eliminate the effect of mutual coupling. The 2D-DOA of uncorrelated signals can be estimated by DOA-matrix method. The parameter pairing between azimuth andelevation is accomplished. Then these estimations are uti-lized to get the mutual coupling coefficients. Based on spatialsmoothing and DOA matrix method, the 2D-DOA of coher-ent signals can be estimated. The Cramer–Rao lower boundis derived at last. Simulation results demonstrate the effec-tiveness and performance of the proposed algorithm.

Keywords VMIMO system · DOA estimation · Multipath ·Mutual coupling

1 Introduction

The demands for higher data rates over longer distances,scarcity of mobile wireless resources, and need for efficiencyof spectrum usage [17,43–45] have motivated the develop-ment of multiple input and multiple output (MIMO) antennaover orthogonal frequency division multiplexing (OFDM)communication systems [1,7,29,32]. MIMO becomes inc-reasingly essential as the demand for broadband wirelessdata transmitter increases. MIMO antenna implementationrequires multiple antennas and multiple-RF chains in boththe receivers and senders sides. However, because the sizeof User Equipments (UEs) is very small, the conventionalMIMO antenna technologies become challenging, especially

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when it comes to placing multiple-RF chains in these devices,which are usually bulky [23]. Due to this, technologies suchas transmit antenna selection [6] and Spatial Modulation [24]have been proposed that exploit the presence of multipleantennas with a single-RF chain. The energy consumption ofsingle-RF chain is much less than that of multiple-RF chains.The Massive MIMO, also known as “large-scale antenna sys-tems” , contains extreme base station and device densities andunprecedented numbers of antennas in fifth generation (5G)system. The energy efficiency problem is the first thing todeal with. As specified in our stated requirements for 5G, theenergy efficiency of the communication chain typically mea-sured in either Joules/bit or bits/Joule will need to improveby about the same amount as the data rate just to maintainthe power consumption. And by more if such consumptionis to be reduces. This implies a several-order-of-magnitudeincrease in energy efficiency, which is extremely challenging.

An alternative approach is that a set of UEs with sin-gle antenna can be grouped together and communicate withthe Base Station (BS) simultaneously on the same resourceblock. The BS equipped with a set of UEs constructs a VirtualMIMO (VMIMO) system. In the VMIMO system, anten-nas at the terminals are virtually jointed together to form aso-called virtual antenna array. The vast proliferation of themobile devices along with the increasing data demands pavethe way for the 5G wireless communication systems. TheVMIMO is believed to be a key technology for 5G mobilecommunications techniques. It enables one to make use of allthe neighboring terminals and amortize the cost of multipleantennas; hence, a large MIMO channel can be created toincrease capacity significantly as well as improve error rateperformance. The use of smart antenna technique is promis-ing to reduce interference, increase coverage and providegeographic information. There are two basic types of smartantennas. One is the fixed multi-beam antenna. While UEis moving, different beams are turned on or off. The othertype is the adaptive array of antennas. In both types of smartantenna, the Direction-of-Arrival (DOA) estimation [14,16]for UEs is crucial.

Multiple Signal Classification (MUSIC) [27,28] and Esti-mation of Signal Parameters via Rotational Invariance Tech-nique (ESPRIT) [26] have been widely implemented in real-time processing [18] for DOA estimation. It is due to thatthe computational complexity is not high and the resolutionof them is acceptable. However, when multipath propagationcaused by reflection and refraction exists, the incident sig-nals may be coherent, thus the conventional DOA estimationalgorithms can not be used. In order to deal with the coher-ent signals, the spatial smoothing technique can be used as apre-processing step [33]. The matrix reconstruction [10,11]and eigenvector singular value decomposition [3,4] are alter-native effective approaches based on the spatial smoothingtechniques.

At present, the DOA estimation of mixed signals (uncor-related and coherent signals) has received considerable atten-tion. Some new differencing methods were proposed to esti-mate the DOA of uncorrelated and coherent signals sepa-rately [21,25,40]. In these methods, the DOA of uncorrelatedsignals has been estimated at first, and then the contributionof the uncorrelated signals has been eliminated for estimat-ing the DOA of coherent signals. However, in some cases,the resulting matrix which is corresponding to the coherentsignals may be canceled completely to be zero [20]. Based onESPRIT and the oblique projection technique, the DOA andmutual coupling coefficients are estimated [37]. In real appli-cation, the 2D-DOA estimation (i.e., elevation and azimuth)of multiple signals is also an important parameter. Based onthe differencing method, the Z-shaped or uniform rectangu-lar planar [41,42] were used for estimating 2D-DOA. How-ever, these methods have the same problem as [21,25,40].The DOAs of uncorrelated signals are first estimated by amodified 2D ESPRIT. Then the contributions of uncorrelatedsignals and noises are eliminated after performing a subtrac-tion operation on the elements of the covariance matrix andonly those of coherent signals remain [35]. Based on prop-agator method, the 2D-DOA estimation of a mixed signalimpinging on a simple structured planar array which consistsof two parallel uniform linear arrays (ULAs) was proposed.However, the computational complexity of the algorithm istremendous because of the spectrum speak searching [30].In addition, the effect of mutual coupling of the 2D-DOAestimation algorithms mentioned above has not been takeninto consideration. Based on the matrix transformation, themutual coupling coefficients are estimated to calibrate theeffect of mutual coupling [12]. Based on the generalizedeigenvalues utilizing signal subspace eigenvectors (GEESE)algorithm for uniform linear array (ULA), the DOA can beaccurately estimated without any calibration sources and themutual coupling coefficients is also proposed [38].

The conventional DOA estimation of UEs is one by one,which is named as sequential scheme. The VMIMO schemeis that the BS estimates the DOA in a parallel way, whichadopts the SA simultaneously. Obviously, when the poweris fixed, the VMIMO scheme is much more energy efficientthan the sequential scheme. The details of energy efficientanalysis is shown in Sect. 2. However, when the UEs com-municate with the BS, the signal may suffer from the multi-path propagation due to the various reflections. It is causedby reflectors and scatters during the course of signal propa-gation. The buildings and hills are the main reflectors whichcause the multipath propagation. The received signals of theBS would be highly correlated or even coherent. Thus thecoherency among the signals is a critical problem that theVMIMO scheme needs to deal with. The DOA estimationalgorithm design is a great challenge for the VMIMO schemewhen the DOA estimation is accomplished. In real applica-

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DOA estimation algorithm for signals in VMIMO systems 95

tions, the DOA estimation of UEs contains two dimensional,i.e., azimuth angle and elevation angle. The 2D-DOA esti-mation is more meaningful. Moreover, the mutual couplingamong the elements would inevitably affect the DOA estima-tion performance. In order to obtain a high resolution of theDOA estimation, the mutual coupling effect has to be elimi-nated or compensated. Thus the 2D-DOA estimation problemin the VMIMO scheme including multipath propagation andmutual coupling has to be solved.

This paper focuses on the 2D-DOA estimation of uncorre-lated and coherent signals in the presence of unknown mutualcoupling in VMIMO systems. The special structure of mutualcoupling matrix (MCM) of ULA is applied to eliminate theeffect of mutual coupling. The 2D-DOA of uncorrelated sig-nals can be estimated by DOA-matrix method. The para-meter pairing of uncorrelated signals between azimuth andelevation is accomplished. Then these estimations are uti-lized to get the mutual coupling coefficients. The obliqueoperator is used to eliminate the information of the uncorre-lated signals. Based on spatial smoothing and DOA matrixmethod, the 2D-DOA of coherent signals can be estimated.Compared with the FBSS-based method, the proposed algo-rithm can estimate more signals. The computational com-plexity of the proposed algorithm is lower than that of theFBSS-based method. The uncorrelated and coherent sig-nals of the proposed algorithm can be carried out in parallelbecause there is no inherent relationship between them. Theprocessing can be further speeded up in practical application.However, the DOA estimation of Automatic Weighted Sub-space Fitting (AWSF) algorithm [5] need multi-dimensionalnonlinear optimization searching, and large numbers ofUEs in the VMIMO system lead to large computationalcost.

The paper is organized as follows. The energy efficientanalysis is given in Sect. 2. The VMIMO system modeland narrowband signal model are introduced in Sect. 3.The DOA estimation of uncorrelated signals is then con-sidered in detail in Sect. 4. In Sect. 5, the mutual cou-pling coefficients estimation is elaborated. The DOA esti-mation of coherent signals is given in Sect. 6. The algo-rithm steps and some concluding remarks for this methodare given in Sect. 7. In Sect. 8, simulation results are shownand discussed to validate the effectiveness of our method.The derivation of Cramer–Rao bound (CRB) is given in theappendix.

2 The energy efficient analysis

Consider a system that a BS communicates with M UEs.The smart antenna technique is used in the BS and onlyone antenna is installed in the UEs. Therefore, the DOAestimation is an important problem to solve in the system.

Generally, there are two methods to estimate the DOA. Thefirst scheme is to estimate the DOAs of UEs one by one,which is named as sequential scheme. The second schemeis that the BS estimates the DOA in a parallel way, which isnamed as the VMIMO scheme. For each time, the numberof the incident signals coming from the UEs is one, then thesequential scheme can exploit most DOA estimation algo-rithms to achieve high estimation accuracy and resolution.It can be seen obviously that the VMIMO scheme is muchmore efficient than the sequential scheme. In order to have anintuitionistic interpretation, an example is given as follows.A snapshot time Tsp is taken to estimate the DOA of one UE.For M UEs, M × Tsp time for snapshots has to be taken forthe sequential scheme and M times DOA estimation have tobe done. The sequential scheme is suitable for the conditionthat the processing capacity of the BS is weak. If the BS hasa strong processing capacity, the VMIMO scheme should beadopted. We assume that the VMIMO scheme can estimateKV DOAs . i.e., the DOAs of KV UEs are estimated simulta-

neously. Then⌈

MKV

⌉times DOA estimation have to be taken

for the BS and the snapshot time is reduced to⌈

MKV

⌉× Tsp.

The power of BS is assumed to be Pt , the energy consump-tion of sequential scheme is Pt ×MTsp. The VMIMO scheme

is Pt

⌈M

KV

⌉Tsp, which takes less energy consumption than

that of sequential scheme. More details can be found in [5].Based on the energy efficient analysis, the VMIMO schemeis adopted. Then the DOA estimation algorithm based on theVMIMO scheme is proposed in the following.

3 The system model and problem formulation

In this section, the VMIMO system is constructed at first.Next, the mathematical model of 2D-DOA estimation foruncorrelated and coherent signals is given, and the array istwo parallel ULAs.

3.1 The VMIMO system model

As shown in Fig. 2, the smart antenna technique is employedin BS for DOA estimation. The BS communicates with MUEs. Based on the smart antenna technique, the DOA esti-mation of UEs becomes more easily. Because of the signalreflection caused by the building, the multipath propagationis taken into consideration. There are two feasible schemesto estimate DOA. One is called the sequential scheme thatthe BS estimates the DOA of the UEs one by one sequen-tially. It is commonly used in the traditional communicationframework. Another is called the VMIMO scheme that a setof UEs with single antenna can be grouped together to com-municate with the BS with multiple antennas on the same

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96 L. Wan et al.

Fig. 1 The sequential scheme and VMIMO scheme

resource block. Then a Virtual MIMO system is structured[17]. These two schemes are shown in Fig. 1.

3.2 The Problem Formulation of DOA Estimation

Consider K narrowband far-field signals impinging on a pla-nar array consisting of two parallel ULAs. Assume that thesmart antenna of the BS is the array antenna with two parallelULAs. The configuration of the two parallel ULAs is shownin Fig. 3.

Each ULA has N identical elements with inter elementspacing dy , and the spacing between two ULAs is dx . Thesub-array 1 and sub-array 2 consist of elements y1 ∼ yN

and elements x1 ∼ xN , respectively. The incident signalsare a mixture of coherent and uncorrelated signals with dis-tinct 2D-DOAs (αk, βk). Assume that there are Kc coher-ent signals undergoing multipath propagation and Ku uncor-related signals undergoing single-path propagation, whereK = Kc + Ku . The coherent signals consist of P groups. Inthe pth group, the coherent signal from

(αpl , βpl

)due to lth

multipath propagation of sp (t) with power σ 2p and the com-

plex fading coefficient is ρpl with∣∣ρpl∣∣ ≤ 1, l = 1, . . . , L p,

p = 1, . . . , P . Then the number of the coherent signals isKc =∑P

p=1 L p, and these coherent signals can be expressedas s1,1 (t) , . . . , s1,L1 (t) , . . . , sP,1 (t) , . . . , sP,L p (t) wherespl (t) = ρpl sp (t). The remaining incident signals arethe uncorrelated ones. Without loss of generally, the signalsKc+1 (t) , . . . , sK (t) are assumed to be the uncorrelated sig-nals. The coherent signals in different groups are uncorrelatedwith each other. Also, the coherent signals are uncorrelatedwith the uncorrelated signals. Then the output of two ULAs(Y (t) and X (t)) can be expressed as

Y (t) =P∑

p=1

L p∑l=1

Ca(αpl , βpl

)ρpl sp (t)

+K∑

k=Kc+1

Ca (αk, βk) sk (t) + N1 (t)

= CAc�sc (t) + CAusu (t) + N1 (t)

= CAEs (t) + N1 (t) (1)

X (t) = CAcDc�sc (t) + CAuDusu (t) + N2 (t)

= CADEs (t) + N2 (t) (2)where N1 (t) and N2 (t) represent the N × 1 additiveGaussian white noise vector with each entry equal to σ 2

n .s (t) is the (P + Ku) × 1 signal vector given by s (t) =[sT

c (t) , sTu (t)]T

, the coherent signal vector is represented assc (t) = [s1 (t) , s2 (t) , . . . , sP (t)]T , the uncorrelated signalvector is represented as su (t) = [sKc+1 (t) , sKc+2 (t) , . . . ,

sK (t)]T , where (·)T denotes transpose. The entries of s (t)are zero mean wide-sense stationary random processes.A = [Ac, Au] represents N × K array manifold matrix,where the coherent signals array manifold matrix is Ac =[Ac1, Ac2, . . . , AcP ], and Acp = [

a(αp1), a(αp2), . . . ,

a(αpL p

)], the uncorrelated signals array manifold matrix is

represented as Au = [a (αKc+1), a(αKc+2

), . . . , a (αK )

].

The steering vector of the coherent signals is expressed

as a(αpk) =

[1, e jτ(αpk), . . . , e j(M−1)τ(αpk)

]T, and the

steering vector of the uncorrelated signals is expressed as

a (αk) = [1, e jτ(αk ), . . . , e j(M−1)τ (αk )]T

. While the matrixE = blkdiag

[�, IKu

], and � = blkdiag

[ρ1, ρ2, . . . , ρP

],

ρ p = [ρp1, ρp2, . . . , ρpL p

]T . The matrix D = blkdiag[Dc, Du], where Dc = blkdiag [Dc1, Dc2, . . . , DcP ], and

Dcp = diag

[e jγ (βp1), e jγ (βp2), . . . , e

jγ(βpL p

)]. The

matrix Du = diag[e jγ (βKc+1), e jγ (βKc+2), . . . , e jγ (βK )

]. In

this paper, diag (·), blkdiag (·) and IKu denote as the diag-onal matrix, block diagonal matrix and Ku × Ku identitymatrix, respectively. τ (α) = 2πdy cos α/λ, and γ (β) =2πdx cos β/λ.

C is the N × N mutual coupling matrix (MCM). Themutual coupling coefficients between two elements areinversely proportional to their distance. The mutual couplingdegree of freedom is assumed to be M0 [8]. When the dis-tance between two elements is larger than (M0 − 1) d, themutual coupling coefficients attenuate to zero. The spacingbetween two ULAs is set to be dx = λ/2. The structure ofthe antenna design is identical, and the elements arrangementof two ULAs is similar. For simplicity, only the mutual cou-pling among elements which belongs to the identical ULAis considered in this paper. Therefore, the MCM of the twoULAs can be assumed to have the same structure. i.e., theentries of the MCMs which belong to two ULAs are iden-tical. The mutual coupling degree of freedom is assumedto be (M0 + 1), which means for the ith element, the cou-pling comes from the (i − M0) th to (i + M0) th elements.The MCM C is constructed as a banded symmetric Toeplitzmatrix, whose first row is c = [1, c1, c2, . . . , cM0 , 0, . . . , 0]satisfying 0 <

∣∣cM0

∣∣ < ∣∣cM0−1∣∣ < · · · < c1 = 1. The MCM

C is given by

123


Fig. 2 DOA estimation for VMIMO scheme considering multipathpropagation

C = T oepli t z(c)

=

⎡⎢⎢⎢⎣

1 c1 · · · cN−1

c1 1 · · · cN−2...

......

...

cN−1 cN−2 · · · 1

⎤⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 c1 · · · cM0 · · · 0

c1 1 c1 · · · . . . 0... c1 1

. . . · · · cM0

cM0 · · · . . .. . . c1

...

0. . . · · · c1 1 c1

0 · · · cM0 · · · c1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3)

Based on the structure of two ULAs, some methods hasbeen proposed without mutual coupling effects [15,19,34].However, it is difficult to estimate 2D-DOA when the mutualcoupling is not known. In the following section, a novel 2D-DOA estimation algorithm with unknown MCM C will beintroduced.

4 2D-DOA estimation of the uncorrelated signals

In order to eliminate the effect of mutual coupling, the firstand last M0 elements of the N = (N − 2M0)-elements ULAare selected as instrumental elements in each ULA [9]. Atthe first glance, it is an unreasonable idea to abandon someinformation that the array receives. However, the effect ofmutual coupling can be eliminated by this sacrifice. Define aselection matrix F = [0(N−2M0)×M0 IN−2M0 0(N−2M0)×M0

],

the output of the center array is expressed as [8]

Y (t) = FY (t)

= FCAc�sc (t) + FCAusu (t) + FN1 (t)

= CAc�sc (t) + CAusu (t) + N1 (t) (4)

where the M × M MCM C of the center array is given by

C = FC =

⎡⎢⎢⎢⎣

cM0 · · · 1 · · · cM0 0 · · · 00 cM0 · · · 1 · · · cM0 · · · 0...

. . .. . . · · · . . . · · · . . .

...

0 · · · 0 cM0 · · · 1 · · · cM0

⎤⎥⎥⎥⎦

(5)

and N1 (t) = FN1 (t). Then an important relationshipbetween the MCM and the steering vector can be obtained,which is represented as [39]

Ca (αk) =

⎡⎢⎢⎢⎢⎢⎣

C1

C2...

CM0−1

CM0−1

⎤⎥⎥⎥⎥⎥⎦

= C

⎡⎢⎢⎢⎣

1vk...

vM0−1k

⎤⎥⎥⎥⎦

= vM0k Da (αk)

= c (αk) a (αk) , (6)

C1 = cM0 + · · · + c1vM0−1k + v

M0k

+ c1vM0+1k + · · · + cM0v

2M0k ,

C2 = cM0vk + · · · + c1vM0k + v

M0+1k

+ c1vM0+2k + · · · + cM0v

2M0+1k ,

CM0−1 = cM0vM−2M0−2k + · · · + c1v

M−M0−2k

+ vM−M0−1k + c1v

M−M0k + · · · + cM0v

M−2k ,

CM0 = cM0vM−2M0−1k + · · · + c1v

M−M0−1k (7)

+ vM−M0k + c1v

M−M0+1k + · · · + cM0v

M−1k ,

C = cM0 + · · · + c1vM0−1k + v

M0k

+ c1vM0+1k + · · · + cM0v

2M0k ,

D = 2M0∑

m0=1

cm0 cos (2m0π cos (αk) d/λ) + 1

where vk = e jτ(αk ) and a (αk) is the ideal steering vector ofthe center array (M0 + 1 ∼ M − M0 elements), and c (αk)

is a scalar function only concerned with the mutual couplingcoefficients and the direction of the incident signal αk , whichis defined as

c (αk) = vM0k D. (8)

When D �= 0, we have CAc� = AcBc� = Ac� andCAu = AuBu . Ac = [a1 (α1) , a2 (α2) , . . . , aKc

(αKc

)]is

the array manifold matrix of the centre array with respectto coherent signals. Bc = diag [B1, B2, . . . , BP ], Bp =diag

[c(αp1), c(αp2), . . . , c

(αpL p

)], and the matrix � =

blkdiag[ρ1, ρ2, . . . , ρP

], where ρp = Bpρ p. Au =[

a(αKc+1

), a(αKc+2

), . . . , a (αK )

]. The matrix Bu =

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98 L. Wan et al.

Fig. 3 The configuration of two parallel ULAs

diag[c(αKc+1

), c(αKc+2

), . . . , c (αK )

]. Equation (4) can

be written in another form as

Y (t) = CAc�sc (t) + CAusu (t) + N1 (t)

= Ac�sc (t) + AuBusu (t) + N1 (t) . (9)

In a similar way, (2) can be written as

X (t) = AcDc�sc (t) + AuBuDusu (t) + N2 (t) (10)

Based on two(N − 1

)× N selection matrices, the centrearray of sub-array 1 and sub-array 2 can be divided into fourarrays, which are called array 1, array 2, array 3 and array 4,respectively.

P1 =⎡⎢⎣

1 · · · 0 0...

. . ....

...

0 · · · 1 0

⎤⎥⎦ , P2 =

⎡⎢⎣

0 1 · · · 0...

.... . .

...

0 0 · · · 1

⎤⎥⎦ (11)

The elements yM0+1 ∼ yN−M0−1, yM0+2 ∼ yM−M0 ,xM0+1 ∼ xN−M0−1 and xM0+2 ∼ xM−M0 elements consti-tute array 1, array 2, array 3, and array 4, respectively, whichcan be expressed as

Y1 (t) = A1c�sc (t) + A1uBusu (t) + N11 (t)

= [A1c�, A1uBu]

s (t) + N11 (t) , (12)

Y2 (t) = [A1cGc�, A1uGuBu]

s (t) + N12 (t) , (13)

X1 (t) = [A1cDc�, A1uBuDu]

s (t) + N21 (t) , (14)

X2 (t) = [A1cGcDc�, A1uGuBuDu]

s (t) + N22 (t) , (15)

where the diagonal matrix G = diag [Gc, Gu], the matrixGc = blkdiag [Gc1, Gc2, . . . , GcP ], and the matrix Gu =diag

[e jτ(αKc+1), e jτ(αKc+2), . . . , e jτ(αK )

].

The auto-covariance matrix of array 1 can be representedas

R1 = E{

Y1 (t) YH1 (t)

}

= [Ac1�, Au1Bu]

Rs[Ac1�, Au1Bu

]H

+ σ 2n IN−1

= Ac1�Rc�H

AHc1 + Au1RuAH

u1 + σ 2n IN−1

= A1ERsEH AH1 + σ 2

n IN−1 (16)

where Ru = BuRuBHu , Rs = blkdiag

[Rc, Ru

]and E =

blkdiag[�, I]. The cross-covariance matrices can be repre-

sented as

R2 = E{

Y2 (t) YH1 (t)

}

= A1GERsEH AH1 + Rn2n1 (17)

R3 = E{

X1 (t) YH1 (t)

}

= A1DERsEH AH1 + Rn3n1 (18)

R4 = E{

X2 (t) YH1 (t)

}

= A1GDERsEH AH1 + Rn4n1 (19)

where the matrix A1 = [Ac1, Au1], and the matrix Rni n1 =[

0(N−2)×1 IN−2

0 01×(N−2)

], i = 1, 2, 3, 4. The DOA matrix is

defined as follows

RDO A1 = R2R†1,

RDO A2 = R3R†1,

RDO A3 = R4R†1. (20)

where (·)† is the pseudo inverse of matrix (·). Based on thenonzero eigenvalue γk of R1 and its corresponding eigenvec-tor uk , R†

1 can be expressed as

R†1 =

P+Ku∑k=1

1

γkukuH

k . (21)

The noise power σ 2n can be estimated by the N − P − Ku

small eigenvalues, and then the noise covariance matricescan be subtracted. The DOA-matrix can be written as

RDO A1 =(

A1GERsEH AH1

) (A1ERsEH AH

1

)†

= JRsEH AH1

×(

A1ERsEH AH1

)†,

J = [A1c1Gc1ρ1, A1c2Gc2ρ2,

. . . , A1cP GcP ρP , Au1Gu]. (22)

According to (16), we have the matrix RsEH AH1 =((

A1E)H (

A1E))−1(

A1E)H

R1. Then (17) can be written

123


as

R2 = A1GE((

A1E)H (

A1E))−1(

A1E)H

R1

+Rn2n1 (23)

Based on Lemma 1 of [41], the relationship among A1,G,RDO A1 and E is given by

RDO A1 A1cpρ p = A1cpGcpρ p, p = 1, 2, . . . , P (24)

RDO A1 A1u = A1uGu . (25)

It can be seen from (25) that Ku diagonal elements of Gu

are a part of the nonzero eigenvalues of RDO A1 . The rank ofRDO A1 is Ku + P , the remaining P eigenvalues are relatedto the D groups of coherent signals from (24). According tothe similar derivation of RDO A1 , RDO A2 and RDO A3 havethe similar property as RDO A1 , which are given by

RDO A2 A2cpρ p = A2cpDcpρ p, p = 1, 2, . . . , P (26)

RDO A2 A2u = A2uDu . (27)

RDO A3 A3cpρ p = A3cpGcpDcpρ p, p = 1, 2, . . . , P (28)

RDO A3 A3u = A3uGuDu . (29)

By taking eigen-decomposition, RDO A1 can be written as

RDO A1 = U1�1U−11 (30)

where �1 = diag[δ11, δ12, . . . , δ1(Ku+P)

]stands for the

eigenvalues of RDO A1 and their corresponding eigenvec-tors are U1 = [

u11, u12, . . . , u1(Ku+P)

]. Because A1cp,

p = 1, 2, . . . , P are Vandermonde matrices, we can notobtain another steering vector based on the combination ofthe steering vectors. The coherent signals do not have thecharacteristic that the uncorrelated signals have, and the char-acteristic is shown in (25). The moduli of the eigenvalueswhich belong to the uncorrelated signals are equal to 1. How-ever, the moduli of the eigenvalues which belong to the coher-ent signals are not equal to 1 [13]. This property can be usedto distinguish the uncorrelated and coherent signals. Basedon (30), RDO A2 and RDO A3 can be written as

RDO A2 = U2�2U−12 (31)

RDO A3 = U3�3U−13 (32)

where the matrix �2 = diag[δ21, δ22, . . . , δ2(Ku+P)

], and

the matrix�3 = diag[δ31, δ32, . . . , δ3(Ku+P)

]are the eigen-

values of RDO A2 and RDO A3 , respectively. The matrixU2 = [

u21, u22, . . . , u2(Ku+P)

]and the matrix U3 =[

u31, u32, . . . , u3(Ku+P)

]are the corresponding eigenvec-

tors. However, when P ≥ 2, the eigen-decomposition ofRDO A2 and RDO A3 are taken separately, thus the parame-ter pairing between δ11, δ12, . . . , δ1(Ku+P) and δ21, δ22, . . . ,

δ2(Ku+P) is a big problem. According to (24)–(29), we can

know that �3 = �2�1. Thus the parameter pairing problemcan be transformed as a problem which minimizes (33)

min∣∣angle (δ3k) − [angle (δ1i ) + angle

(δ2 j)]∣∣ (33)

Then parameter pairing is accomplished. The 2D-DOA ofthe uncorrelated signals can be represented as

αk = arccos

(λ

2πdy

[angle (δ1k) + angle

(δ3k

δ2k

)]),

βk = arccos

(λ

2πdx[angle (δ2i )] + angle

(δ3k

δ1k

)),

k = 1, 2, . . . , Ku (34)

5 The Mutual coupling coefficients estimation

Based on the 2D-DOA estimation of the uncorrelated sig-nals, the estimations of mutual coupling coefficients can beobtained. The covariance matrix of Y (t) can be expressed as

RY = E{

Y (t) YH (t)}

= CAERsEH AH CH + σ 2n IN

= CAc�Rc�H AH

c CH + CAuRuAHu CH

+ σ 2n IN . (35)

The rank of CAERsEH AH CH is Ku + P . Take theeigen-decomposition of RY , the Ku + P big eigenvaluesλ1, λ2, . . . , λKu+P and K − (Ku + P) small eigenvaluesλKu+P+1 = λKu+P+2 = · · · = λK = σ 2

n can beobtained, respectively. Their corresponding eigenvectors areu1, u2, . . . , uM . The signal subspace is spanned by CAc

and CAu jointly, which is orthogonal to the noise subspacespanned by uKu+P+1, uKu+P+2, . . . , uM

UHn CAcpρ p = 0, p = 1, 2, . . . , P

UHn Ca (αk) = 0, k = 1, 2, . . . , Ku . (36)

By considering the complex symmetric Toeplitz form of C,Ca (α) can be expressed as [22]

Ca (α) =

⎡⎢⎢⎢⎣

1 c1 · · · cN−1

c1 1 · · · cN−2...

......

...

cN−1 cN−2 · · · 1

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎣

1e jτ(α)

...

e j(N−1)τ (α)

⎤⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

1 e jτ(α)

e jτ(α) 1 + e j2τ(α)

e j2τ(α) e jτ(αk ) + e j3τ(α)

......

e j(N−2)τ (α) e j(N−3)τ (α) + e j(N−1)τ (α)

e j(N−1)τ (α) e j(N−2)τ (α)

123

100 L. Wan et al.

· · · e j(N−2)τ (α) e j(N−1)τ (α)

· · · e j(N−1)τ (α) 0· · · 0 0. . .

... 0· · · 1 0· · · e jτ(α) 1

⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎣

1c1...

cN−1

⎤⎥⎥⎥⎦

= T (α) c (37)

where T (α) is the sum of two N × (M0 + 1) [12].Based on (35) and the estimations of α1, α2, . . . , αKu , we

have

UHn T (αk) c = 0, k = 1, 2, . . . , Ku (38)

It can be seen that (38) is the linear equations of mutualcoupling coefficients c. The coefficient matrix can be definedas [37]

Q =⎡⎢⎣

UHn T (α1)

...

UHn T(αKu

)

⎤⎥⎦ . (39)

Then (38) can be written as

Qc = 0, (40)

where Q is a Ku (N − Ku − P) × (M0 + 1) matrix, Q =[q1, q2, . . . , qKu

]. Due to c(1) = 1, we have

Qc = [q1, q2, . . . , qM0+1]⎡⎢⎢⎢⎣

1c1...

cM0

⎤⎥⎥⎥⎦ = 0. (41)

When Ku (N − Ku − P) ≥ M0−1, the least square solutionis obtained as [39]

[c1, c2, . . . , cM0

] = −[q2, . . . , qM0+1]†q1. (42)

The mutual coupling coefficients estimation is completed.

6 2D-DOA estimation of the coherent signals

In order to estimate the DOA of coherent signals, the infor-mation of uncorrelated signals has to be eliminated from thecovariance matrix of the array output. The DOA estimation ofcoherent signals can be accomplished by the reduced array.However, the loss of array aperture exists. In order to solvethis problem, the original array is used here. The uncorre-lated signal steering matrix Au and the MCM C can be con-structed by the estimates of α1, . . . , αKu and c1, c2, . . . , cM0 ,

respectively. Based on the oblique projection technique, theinformation of coherent signals can be separated from uncor-related signals. Then an N × N oblique projection operatorP can be defined as [37]

P = CAu

(AH

u CH P⊥CAc�

CAu

)−1AH

u CH P⊥CAc�

(43)

where P⊥CAc�

stands for the orthogonal complement spacespanned by the column of CAc�. The oblique projectionoperator P is difficult to obtain without Ac. Then an alterna-tive method for computation of oblique projector is used here[30]. The cross-covariance matrix between Y (t) and X (t) isexpressed as

RXY = E{

X (t) YH (t)}

= CAcDc�Rc�H AH

c CH

+ CAuDuRuAHu CH . (44)

Then the oblique projector computation method for RXY isintroduced as follows. A new N × N matrix is defined as

Re = RXY P⊥CAu

= CAcDc�Rc�H AH

c CH P⊥CAu

(45)

where P⊥CAu

= IN − CAu((CAu)H CAu

)−1(CAu)H . Obvi-

ously the rank of the matrix Re is P, its QR decompositionis expressed as

Re� = QR

= [q1, q2, . . . , qN] [ R1

0(N−P)×N

]= Q1R1 (46)

where the matrix Q1 = [q1, q2, . . . , qP], the matrix Q2 =[

qP+1, qP+2, . . . , qN]

and Q = [Q1 Q2

]is a N × N

unitary matrix. R1 is a P × N full row rank matrix and �

is a permutation matrix. Referring to the conclusion in [30],the alternative computation of the oblique projector can beobtained as follows

PXY = CAu

(Q2QH

2 CAu

)†. (47)

Then the matrix HXY that does not contain the informationof the uncorrelated signals is expressed as

HXY = (I − PXY ) RXY

= CAcDc�Rc�H AH

c CH (48)

Another method of the oblique projector computation forRY is introduced as follows. Referring to the derivation in[36], when Ku + P ≤ N , the oblique projector computationmethod is given by

PY = CAu

(AH

u CH R†Y CAu

)−1AH

u CH R†Y . (49)

123


Similar as HXY , HY is expressed as

HY = (I − PY ) RY = CAc�Rc�H AH

c CH . (50)

In order to eliminate the effect of mutual coupling, we canmultiply C−1 on both sides of HXY and HY . Then RXY andRY can be obtained which are expressed as

RXY = C−1RXY

(C−1)H = AcDc�Rc�

H AHc , (51)

RY = C−1RY

(C−1)H = Ac�Rc�

H AHc . (52)

Based on (51) and (52), the spatial smoothing is performedon RXY and RY , respectively. Notice that the smoothing hereis only based on mathematical submatrices and not on realsubarrays. The size of subarrays is assumed to be M1, and thenumber of subarrays is L = N −M1+1. The submatrices canbe equivalent to the covariance matrices of some overlappedsubarrays, which are expressed as

RlXY = RXY (l : l + N − 1, l : l + N − 1) , (53)

RlY = RY (l : l + N − 1, l : l + N − 1) . (54)

The smoothed auto-covariance matrix RlY can be obtained

RY = 1

N − M1 + 1

N−M1+1∑l=1

RlY = Asub1RsAH

sub1 (55)

where Asub1 is the array manifold matrix of the first equiv-alent subarray and Rs is covariance matrix of the coherentsignals, which is given by

Rs = 1

N − M1 + 1

N−M1+1∑l=1

Gc�Rc�H GH

c . (56)

The smoothed cross-covariance matrix RlXY can be obtained

RXY = 1

N − M1 + 1

N−M1+1∑l=1

RlXY

= Asub1DcRsAHsub1. (57)

The DOA-matrix is redefined as follows

RDO A = RXY(RY)†

. (58)

The eigen-decomposition is taken on RDO A, we have

RDO A = Uc�cU−1c , (59)

where �c = diag[δc1, δc2, . . . , δcKu

]stands for the eigen-

values of RDO A and their corresponding eigenvectors areUc = [uc1, uc2, . . . , ucKu

]. Then α1, . . . , αKc and β1, . . . ,

βKc can be obtained.

αk = arccos T

T =(

λ2πdx (M1−1)

M1∑i=2

1i−1

[angle

∣∣∣ uck (i)uck (1)

∣∣∣])

,

βk = arccos

(λ

2πdyangle (δck)

), k = 1, 2, . . . , Kc (60)

where uck (i) stands for the ith element of the eigenvectoruck .

When the 2D-DOAs of all the incident signals areobtained, they can also be converted to the azimuth andelevation as φ = arctan (cos β/cos α) and ϕ = arccos√

(cos α)2 + (cos β)2, respectively.

7 Summary of the algorithm and discussions

In this section, the steps of the proposed algorithm are sum-marized at first. Then the minimum acquired number of ele-ments is analyzed. The steps of the proposed algorithm aresummarized as follows.

Step 1: Based on the selection matrix F, the effect of MCMis eliminated. The output of the center array are expressed asY (t) and X (t), respectively.

Step 2: The centre array of sub-array 1 and sub-array 2are divided into four arrays. The output of these arrays areY1 (t) Y2 (t) and X1 (t), X2 (t), t = 1, 2, . . . , Np which arecalled array 1, array 2, array 3 and array 4, respectively. Np

is the snapshot number. The estimate of R1 is expressed as

R1 = 1

Np

Np∑t=1

Y1 (t) Y1 (t) H . (61)

Then the estimates of R2, R3, R4 can be obtained in the sim-ilar way. The estimates RDO A1 , RDO A2 and RDO A3 of DOAmatrix RDO A1 , RDO A2 and RDO A3 can be obtained respec-tively.

Step 3: The eigen decomposition is taken on RDO A1 ,RDO A2 and RDO A3 , respectively. The parameter pairingbetween azimuth and elevation can be accomplished based on(33). The 2D-DOA of uncorrelated signals can be separatedfrom the mixed signals by the judgment criterion in Section 3.Based on (34), the 2D-DOA estimation of the uncorrelatedsignals is completed.

Step 4: The orthogonality between signal subspace andnoise subspace is used for mutual coupling coefficients esti-mation. In order to construct the linear equations (40) ofmutual coupling coefficients c, the form of Ca (α) is changed.Based on (42), the mutual coupling coefficients estimation isaccomplished.

Step 5: Based on (47) and (49), the oblique projectiontechnique is used. the information of coherent signals can beseparated from uncorrelated signals.

123

102 L. Wan et al.

Step 6: Perform eigen-decomposition on RDO A (59) toresolve the coherent signals as in (60).

As mentioned above, the 2D-DOAs of uncorrelated andcoherent signals are estimated separately. Then the uncorre-lated and the coherent signals coming from the same directioncan be distinguished by this two stage process. More signalsthan array sensors can be resolved. The proposed algorithmcan estimate 2D-DOA in parallel, which is suitable for prac-tical application. In the following, the required number ofelements for DOA estimation will be analyzed.

In order to eliminate the effect of mutual coupling, onlyN − 2M0 elements can be used for estimating uncorre-lated signals. The sub-array 1 and sub-array 2 both con-tain N − 2M0 − 1 elements. The number of sub-arrayhas to be satisfied N − 2M0 − 1 ≥ Ku + P + 1. Thereneeds N ≥ Ku + P + 2M0 + 2 to estimate uncorre-lated signals. In order to estimate mutual coupling coeffi-cients, Ku (N − Ku − P) ≥ M0 − 1 is necessary. Thus

N ≥ Ku+P+⌈

M0−1Ku

⌉. In order to estimate coherent signals,

N − L +1 ≥ Kc +1 and L ≥ max (L1, L2, . . . L P ) = Lmax

are needed. N ≥ Kc + Lmax has to be satisfied. Thusthe required number of elements for DOA estimation isN ≥ max (Ku + P + 2M0 + 2, Kc + Lmax).

There is another phenomenon called “blind angles”, whichmeans the array is blind which can not receive any incidentsignal from some particular angles [39]. These angles makeD = 0. Then whenever there is one incident signal with itsDOA α j = αk, k = 1, 2, . . . , Ku . , the array output of thecenter N − 2M0 elements can be represented as

Y (t) =P∑

p=1

L p∑l=1

Ca(αpl , βpl

)ρpl sp (t)

+K∑

k=Kc+1


=P∑

p=1

L p∑l=1

Ca(αpl , βpl

)ρpl sp (t)

+K∑

k=Kc+1, j �=k


The equation above implies that the jth incident signal isnot contained. So at some particular angles, the array is blindwhich can not receive any incident signal from these direc-tions. Thus the estimated angles of the uncorrelated signalswould be lost. The number of the lost angles can be estimatedby the difference in the large eigenvalues between RY andR1. The number of the lost angles is assumed to be K0, soonly Ku−K0 uncorrelated signals are estimated. Fortunately,the lost angle can be found back in the estimation of coherent

signals. When N ≥ Ku + P +⌈

M0−1Ku−K0

⌉, the uncorrelated

signal steering matrix, the MCM and the oblique projectionoperator P can be constructed by the Ku − K0 estimators.The remaining K0 DOA of uncorrelated signals will be con-tained in (56). The K0 lost angles of uncorrelated signals andKc coherent signals can be estimated together [37].

8 The simulation results

In this section, simulations are shown to demonstrated prop-erties of the proposed algorithm. The method of FBSS pro-posed in [31] combined with the DOA-matrix method andCRB are used as the comparison. Each ULA of the parallel-shaped array has twelve elements. The spacing of elementsdy and the spacing between two ULAs dx are equivalent withhalf-wavelength (dy = dx = λ/2). 500 Monte Carlo trialsare taken from the simulation. The average root mean squareerror (RMSE) is defined as

RMSE =√√√√ 1

500I

500∑m=1

I∑k=1

(αk (m) − αk

)2 +(βk (m) − βk

)2(62)

RMSEc =√√√√ 1

500 ‖c1‖500∑

m=1

∥∥c1 (m) − c1∥∥× 100 % (63)

where αk (m), βk (m) and c1 (m) are the estimates of αk , βk

and c1 of the mth Monte Carlo trials, respectively. I is thenumber of all the uncorrelated signals or all the coherent sig-nals. ‖·‖ is defined as the Frobenius norm. The uncorrelatedand coherent signals of the proposed algorithm can be esti-mated separately. In this comparison, the FBSS method hasto estimate the uncorrelated and coherent signals together,the same method of eliminating the mutual coupling is used.

Five incident signals with equal power come from thedirection (100◦, 60◦), (75◦, 30◦), (86◦, 53◦), (76◦, 67◦),(16◦, 83◦), and without loss of generality the first two sig-nals are coherent signals, and the fading coefficients are[1, 0.7415 − j0.5100

]. The last three signals are uncorrelated

signals. The mutual coupling coefficients are[1, j0.4499 − 0

.5362]. In Sect. 5, the size of subarrays is M1 = 7. The num-ber of snapshot is 500. The RMSE of the DOA estimation ofuncorrelated and coherent signals versus input SNR is shownin Figs. 4 and 5, respectively.

For the uncorrelated signals, it can be seen from Fig. 4that the DOA estimation errors of the two algorithms reducewith the SNR increases. The DOA estimation performanceof proposed algorithm outperforms than that of the FBSSalgorithm. The azimuth estimation RMSE of the proposedalgorithm approximates to the CRB at low SNR. Both thetwo algorithms approximate to the CRB at high SNR. Forthe coherent signals, it can be seen from Fig. 5 that theRMSE of FBSS algorithm changes very little with the varying

123


−10 −5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

SNR(dB)

RM

SE

(deg

)FBSS elevation proposed elevation FBSS azimuth proposed azimuthCRB elevationCRB azimuth

Fig. 4 RMSE of DOA estimations versus input SNR for uncorrelatedsignal

−10 −5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

SNR(dB)

RM

SE

(deg

)

proposed elevationproposed azimuthFBSS elevationFBSS azimuthCRB elevationCRB azimuth

Fig. 5 RMSE of DOA estimations versus input SNR for coherent sig-nal

SNR. However, the RMSE of the proposed algorithm reduceswith SNR increases. The RMSE of the proposed algorithmapproximates to the CRB at high SNR. This result is mainlycaused that the proposed algorithm estimates the uncorre-lated signals and the coherent signals separately. However,the FBSS algorithm estimates both the signals simultane-ously. The mutual coupling estimation versus input SNR isshown in Fig. 6. The results show that the mutual coupling haslittle influence on the proposed algorithm. Although somearray apertures are used to eliminate the effect of mutualcoupling, the proposed algorithm still has high estimationaccuracy and approaches to CRB at high SNR.

9 Conclusions

In this paper, the VMIMO scheme for DOA estimation isintroduced which replaces the sequential scheme. It means

−10 −5 0 5 10 15 200

0.5

1

1.5

2

2.5

SNR(dB)

RM

SE

(%)

proposed methodCRB

Fig. 6 RMSE of the mutual coupling coefficients versus input SNR

that the BS can detect the 2D-DOA of a set of UEs togethersimultaneously. Based on the VMIMO scheme, a novel 2D-DOA estimation algorithm in the scenario that uncorre-lated signals and coherent signals coexist in the presenceof unknown mutual coupling is proposed. The 2D-DOA ofuncorrelated signals and coherent signals can be estimated bythe DOA matrix method. The mutual coupling coefficientsare estimated without the help of any calibration sourcesat known locations. Although the “blind angles” phenom-enon may be happen with some particular angles. Fortu-nately, the lost angle can be found back in the estimationof coherent signals. Simulation results validate the effec-tiveness of the proposed method. The proposed algorithmcan estimate 2D-DOA in parallel, which is suitable for prac-tical application. Compared with the traditional sequentialscheme, the VMIMO scheme can save more energy, whichwill have a prosperous application in 5G wireless communi-cation systems.

Acknowledgments This work was supported in part by the QingLan Project, the National Science Foundation of China under Grant61201410 and 61401147, the Natural Science Foundation of JiangSuProvince of China,No.BK20140248, the Fundamental Research Fundsfor the Central Universities (Program No. HEUCF140803).This workhas been partially supported by Instituto de Telecomunicações, NextGeneration Networks and Applications Group (NetGNA), Covilhã Del-egation, by Government of Russian Federation, Grant 074-U01, and byNational Funding from the FCT - Fundação para a Ciência e a Tec-nologia through the Pest-OE/EEI/LA0008/2013 Project.

Appendix

The Cramer–Rao lower bound (CRB) of 1D-DOA estimationin the presence of mutual coupling is given in [12,37]. In thisAppendix, the CRB of joint 2D-DOA and mutual couplingestimation is derived. Consider the array output vector x (t)

123

104 L. Wan et al.

as a complex Gaussian vector with zero mean. Define A Δ=[Ac, Au], E Δ= blkdiag

[�, IKu

], the covariance matrix of

x (t) is expressed as

Rx = E{

x (t) x(t)H}

= CAERsEH AH CH (64)

For L statistically independent observations of x (t), the loga-rithm of the likelihood function (the joint probability densityfunction, PDF) can be written as [2,38]

� = ln { f (x (1) , x (2) , . . . , x (L))}= const − L · ln {det {Rx }}

−L∑

t=1

x(t)H R−1x x (t)

= const − L · ln {det {Rx }}−L · tr

{R−1

x Rx

}(65)

where the estimate of auto-correlation covariance Rx is givenby

Rx = 1

L

L∑l=1

x (l) x(l)H . (66)

The unknown parameter vector of Rx is defined as

η =[αT ,βT ,μT , νT , κT , ςT

]T

α = [α11, . . . , α1L1 , . . . , αP1 , . . . , αP L P ,

αKc+1, . . . , αK]T

β = [β11, . . . , β1L1 , . . . , βP1 , . . . , βP L P ,

βKc+1, . . . , βK]T (67)

In order to obtain the unique CRB of ρk , the fadingcoefficient of the signal in the pth group is normalized tounity. For the sake of simplify, αp1 is assumed to be thesmallest DOA in the pth group in the following deriva-tion. μ = [μ12, . . . , μ1L1 , . . . , μP2 , . . . , μP L P

]T and ν =[ν12, . . . , ν1L1 , . . . , νP2 , . . . , νP L P

]T are defined as the real

part and = [ρ1(2 : P1)T , . . . , ρD(2 : PD)T ]T is defined

as the imaginary part, respectively. κ and ς are defined asthe real part and imaginary part of c1, respectively. The kthelement in a vector, for example, η is defined as ηk . Then thegeneral expression of the (m, n)th element in Fisher infor-mation matrix (FIM) can be expressed as

Fηmηn = −E

{∂2Θ

∂ηm∂ηn

}(68)

Based on the relationship

∂R−1x

∂ηm= −R−1

x∂Rx

∂ηmR−1

x (69)

∂ ln {det {Rx }}∂ηm

= tr

{R−1

x∂Rx

∂ηm

}(70)

the first derivative of Θ is obtained

∂Θ

∂ηm= −Ltr

{R−1

x∂Rx

∂ηm

}

+ Ltr

{R−1

x∂Rx

∂ηmR−1

x Rx

}

= Ltr

{R−1

x∂Rx

∂ηm

(R−1

x Rx − I)}

. (71)

The second derivative of Θ is given by

∂Θ

∂ηmηn= Ltr

{R−1

x∂Rx

∂ηm

(R−1

x Rx − I)}

= Ltr

{(∂

(R−1

x∂Rx

∂ηm

)/∂ηn

)(R−1

x Rx − I)}

+ Ltr

{R−1

x∂Rx

∂ηm

∂(R−1

x Rx − I)

∂ηn

}

= Ltr

{(∂

(R−1

x∂Rx

∂ηm

)/∂ηn

)(R−1

x Rx − I)}

− Ltr

{R−1

x∂Rx

∂ηm

(R−1

x∂Rx

∂ηnR−1

x Rx

)}. (72)

Due to E{Rx} = Rx , the expectation of both sides of (70)

is taken, we have

Fηmηn = Ltr

{R−1

x∂Rx

∂ηmR−1

x∂Rx

∂ηn

}. (73)

In the subsequent derivation process, the FIM expression ofmixed signals is derived. The notation Rηm defines ∂R/∂ηm .

Derivatives with respect to DOA

Based on the expression of covariance matrix (64), the partialderivative of Rx with respect to the mth element αm of α canbe written as

∂Rx

∂αm= CAαm ERsEH AH CH

+ CAERsEH AHαm

CH . (74)

Based on tr{R + RH

} = 2Re {tr {R}}, we have

Fαmαn = Ltr

{R−1

x∂Rx

∂αmR−1

x∂Rx

∂αn

}

= Ltr{

R−1x

(CAαm ERsEH AH CH

+ CAERsEH AHαm

CH)

× R−1x

123


(CAαn ERsEH AH CH

+ CAERsEH AHαn

CH)}

= 2LRe{

tr{

R−1x CAαm ERsEH AH

× CH R−1x CAαn ERsEH AH CH

+ R−1x CAERsEH AH

αm

× CH R−1x CAαn ERsEH AH CH

}}(75)

Since only the mth column of Aαm is nonzero, then Aαm can be

represented as Aαm = Aαγ mK

(γ m

K

)T , where the mth columnof the identity matrix is defined as γ m

K . Aα is the derivativematrix of the array manifold matrix, which is expressed as

Aα =[

da (α11)

dα11, . . . ,

da(α1P1

)

dα1P1

, . . . ,da (αD1)

dαD1,

. . . ,da(αD PD

)

dαD PD

,da(αKc+1

)

dαKc+1, . . . ,

da (αK )

dαK

]T

(76)

Then (73) can be written as

Fαmαn = 2LRe {tr{

R−1x CAαγ m

K

(γ m

K

)T ERsEH AH

× CH R−1x CAαγ n

K

(γ n

K

)T ERsEH AH CH

+ R−1x CAERsEH γ m

K

(γ m

K

)T AHα ×

CH R−1x CAαγ n

K

(γ n

K

)T ERsEH AH CH}}

= 2LRe{((

γ mK

)T ERsEH AH CH R−1x CAαγ n

K

)

×((

γ nK

)T ERsEH AH CH R−1x CAαγ m

K

)

+((

γ mK

)T AHα CH R−1

x CAαγ nK

)×

((γ n

K

)T ERsEH AH CH R−1x CAERsEH γ m

K

)}

(77)

Then the FIM that corresponds to α can be expressed as

Fαα = 2LRe{(

ERsEH AH CH R−1x CAα

)

(


)T

+(

AHα CH R−1

x CAα

)

(

ERsEH AH CH R−1x CAERsEH

)T}

(78)

where denotes the Hadamard product. Similarly, the FIMthat corresponds to β can be expressed as

Fββ = 2LRe{(

ERsEH AH CH R−1x CAβ

)

(


)T

+(

AHβ CH R−1

x CAβ

)

(


)T}

(79)

where

Aα =[

da (β11)

dβ11, . . . ,

da(β1P1

)

dβ1P1

, . . . ,da (βD1)

dβD1,

. . . ,da(βD PD

)

dβD PD

,da(βKc+1

)

dβKc+1, . . . ,

da (βK )

dβK

]T

.

(80)

The FIM that corresponds to the cross terms between α andβ is

Fαβ = 2LRe{(


)

(


)T

+(

AHα CH R−1

x CAβ

)

(


)T}

(81)

Derivatives with respect to fading coefficients

Ac = [Ac1 (1 : N , 2 : L1) , . . . , AcD (1 : N , 2 : L P )] , and� = blkdiag

{ρ1 (2 : L1) , . . . , ρD (2 : L P )

}. The matrix

�r = blkdiag{1(L1−1)×1, . . . , 1(L P−1)×1

}and � i =

blkdiag{j(L1−1)×1, . . . , j(L P−1)×1

}, where all the elements

of the vector 1 are equal to 1 and all the elements of the vec-tor j are equal to the imaginary unit j . According to (73), the(rm, rn)th element of the FIM with respect to fading coeffi-cients can be expressed as

Fμmμn = Ltr

{R−1

x∂Rx

∂μmR−1

x∂Rx

∂μn

}

= Ltr{

R−1x

(CAEμm RsEH AH CH

+ CAERsEHμm

AH CH)

× R−1x

(CAEμn RsEH AH CH

+ CAERsEHμn

AH CH)}

= Ltr{

R−1x

(CAc�μm Rcs�

H AHc CH

+ CAc�Rcs�H

μmA

Hc CH

)

× R−1x

(CAc�μn Rcs�

H AHc CH

+ CAc�Rcs�H

μnA

Hc CH

)}

= 2LRe{

tr{

R−1x CAc�μm Rcs�

H AHc

123

106 L. Wan et al.

× CH R−1x CAc�μn Rcs�

H AHc CH

+ R−1x CAc�Rcs�

H

μmA

Hc

CH R−1x CAc�μn Rcs�

H AHc CH

}}(82)

Based on �μm = γmKc−P

(γm

Kc−P

)T�r, the real part of fading

coefficients of the FIM can be represented as

Fμμ = 2LRe{

R−1x CAc�μm Rcs�

H AHc


H AHc CH

+ R−1x CAc�Rcs�

H

μmA

Hc


H AHc CH

}

= 2LRe{(

�rRcs�H AH

c CH R−1x CAc

)

(�rRcs�

H AHc CH R−1

x CAc

)T

+(

AHc CH R−1

x CAc

)

(�rRcs�

H AHc CH R−1

x CAc�Rcs�Hr

)T}

(83)

Based on �νm = γ mKc−P

(γ m

Kc−P

)T� i, the imaginary part

of fading coefficients of the FIM can be represented as

Fνν = 2LRe{(

� iRcs�H AH

c CH R−1x CAc

)

(� iRcs�

H AHc CH R−1

x CAc

)T

+(

AHc CH R−1

x CAc

)

(� iRcs�

H AHc CH R−1

x CAc�Rcs�Hi

)T}

(84)

The FIM that corresponds to the cross terms between μ andν is

Fμν = 2LRe{(

�rRcs�H AH

c CH R−1x CAc

)

(� iRcs�

H AHc CH R−1

x CAc

)T

+(

AHc CH R−1

x CAc

)

(� iRcs�

H AHc CH R−1

x CAc�Rcs�Hr

)T}

(85)

Derivatives with respect to mutual coupling coefficients

Based on (73), the mth and nth element of Fκκ , Fςς and Fκς

can be given respectively as follows

Fκmκn = 2LRe{

tr{

R−1x Cκm AERsEH AH

× CH R−1x Cκn AERsEH AH CH

+ R−1x Cκm AERsEH AH

× CH R−1x CAERsEH AH CH

κn

}}(86)

Fςmςn = 2LRe{

tr{

R−1x Cςm AERsEH AH

× CH R−1x Cςn AERsEH AH CH

+ R−1x Cςm AERsEH AH


ςn

}}(87)

Fκmςn = 2LRe{

tr{

R−1x Cκm AERsEH AH

× CH R−1x Cςn AERsEH AH CH

+ R−1x Cκm AERsEH AH


ςn

}}(88)

where

Cκm = toepli t z{[

0,(γ m

M0

)T, 01,(M−M0−1)

]},

m = 1, . . . , M0. (89)

toepli t z {z} stands for the symmetric Toeplitz matrix con-structed by the vector z.

DOA-fading coefficients cross terms

Fαμ = 2LRe{(

ERsEH AH CH R−1x CAc

)

(�rRcs�

H AHc CH R−1

x CAα

)T

+(

AHα CH R−1

x CAc

)

(�rRcs�

H AHc CH R−1

x CAERsEH)T}

(90)

Fαν = 2LRe{(


)

(� iRcs�

H AHc CH R−1

x CAα

)T

+(

AHα CH R−1

x CAc

)

(� iRcs�

H AHc CH R−1

x CAERsEH)T}

(91)

Fβμ = 2LRe{(


)

(�rRcs�

H AHc CH R−1

x CAβ

)T

+(

AHβ CH R−1

x CAc

)

(�rRcs�

H AHc CH R−1

x CAERsEH)T}

(92)

Fβν = 2LRe{(


)

(� iRcs�

H AHc CH R−1

x CAβ

)T

+(

AHβ CH R−1

x CAc

)

123


(� iRcs�

H AHc CH R−1

x CAERsEH)T}

(93)

DOA-mutual coupling coefficients cross terms

Fακn = 2LRe{

diag(

ERsEH AH CH R−1x Cκn

× AERsEH AH CH R−1x CAα

)

+ diag(

ERsEH AH CH R−1x C

× AERsEH AH CHκn

R−1x CAα

)}(94)

Fαςn = 2LRe{

diag(

ERsEH AH CH R−1x Cςn

× AERsEH AH CH R−1x CAα

)

+ diag(


× AERsEH AH CHςn

R−1x CAα

)}(95)

Fβκn = 2LRe {diag(

ERsEH AH CH R−1x Cκn

× AERsEH AH CH R−1x CAβ

)

+ diag(

ERsEH AH CH R−1x C×

AERsEH AH CHκn

R−1x CAβ

)}(96)

Fβςn= 2LRe {diag

(ERsEH AH CH R−1

x Cςn

× AERsEH AH CH R−1x CAβ

)

+ diag(


× AERsEH AH CHςn

R−1x CAβ

)}(97)

where diag (A) is a column vector constructed by the maindiagonal elements of matrix A.

Fading coefficients-mutual coupling coefficients cross terms

Fμκn = 2LRe {diag(�rRcs�

H AHc CH R−1

x

× Cκn AERsEH AH CH R−1x CAc

)

+diag(�rRcs�

H AHc CH R−1

x C

× AERsEH AH CHκn

R−1x CAc

)}(98)

Fμςn = 2LRe {diag(�rRcs�

H AHc CH R−1

x

× Cςn AERsEH AH CH R−1x CAc

)

+ diag(�rRcs�

H AHc CH R−1

x C

× AERsEH AH CHςn

R−1x CAc

)}(99)

Fνκn = 2LRe {diag(� iRcs�

H AHc CH R−1

x

× Cκn AERsEH AH CH R−1x CAc

)

+ diag(� iRcs�

H AHc CH R−1

x

× CAERsEH AH CHκn

R−1x CAc

)}(100)

Fνςn = 2LRe {diag(� iRcs�

H AHc CH R−1

x

×Cςn AERsEH AH CH R−1x CAc

+ diag(� iRcs�

H AHc CH R−1

x C

× AERsEH AH CHςn

R−1x CAc

)}(101)

Based on the above formulations, the whole FIM can beexpressed as

Fηη =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

Fαα Fαβ Fαμ Fαν Fακ Fας

FTαβ Fββ Fβμ Fβν Fβκ Fβς

FTαμ FT

βμ Fμμ Fμν Fμκ Fμς

FTαν FT

βν FTμν Fνν Fνκ Fνς

FTακ FT

βκ FTμκ FT

νκ Fκκ Fκς

FTας FT

βς FTμς FT

νς FTκς Fςς

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(102)

Define H = F−1ηη , the CRBs of coherent signals, uncorre-

lated signals and mutual coupling coefficients can be given,respectively, as

C RB�c =√√√√ 1

2Kc

( Kc∑k=1

Hkk +K+Kc∑

k=K+1

Hkk

)(103)

C RB�u = √√√√√ 1

2Kc

⎛⎝

K∑k=Kc+1

Hkk +2K∑

k=K+Kc+1

Hkk

⎞⎠ (104)

C RBc =

√√√√√ 1

‖c1‖

⎛⎝

2(K+Kc−P+M0)∑k=2(K+Kc−P)

Hkk

⎞⎠ (105)

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Liangtian Wan received theB.S. degree in Electrical andInformation Engineering fromHarbin Engineering University,China in 2011. He is cur-rently working toward his M.S.and PhD degree in College ofInformation and CommunicationEngineering at Harbin Engineer-ing University. He has servedas a reviewer of more than10 journals. His research inter-ests include array signal process-ing, compressed sensing and itsapplications.

Guangjie Han is currently a Pro-fessor of Department of Infor-mation & Communication Sys-tem at Hohai University, China.He is also a visiting researchscholar of Osaka University fromOct. 2010 to Oct. 2011. He fin-ished the work as a post doc-tor of Department of ComputerScience at Chonnam NationalUniversity, Korea, in February2008. He worked in ZTE Com-pany from 2004 to 2006, wherehe held the position of ProductManager. He received his Ph.D.

degree in Department of Computer Science from Northeastern Uni-versity, Shenyang, China, in 2004. He has published over 120 papersin related international conferences and journals. He has served in theeditorial board of up to 14 international journals, including Journal ofInternet Technology and KSII Transactions on Internet and InformationSystems. He has served as a Co-chair for more than 20 international con-ferences/workshops; a TPC member of more than 50 conferences. Heholds 49 patents. He has served as a reviewer of more than 50 jour-nals. He had been awarded the Best Paper Awards of the ComManTel2014 and the Chinacom 2014. His current research interests are Sen-sor Networks, Computer Communications, Mobile Cloud Computing,Multimedia Communication and Security. He is a member of IEEE andACM.

Joel J. P. C. Rodrigues is aprofessor in the Department ofInformatics of the University ofBeira Interior, Covilhã, Portu-gal, and researcher at the Insti-tuto de Telecomunicações, Por-tugal. He received a Habilita-tion from the University of HauteAlsace, France, a PhD degree ininformatics engineering, an MScdegree from the University ofBeira Interior, and a five-yearBSc degree (licentiate) in infor-matics engineering from the Uni-versity of Coimbra, Portugal. His

main research interests include sensor networks, e-health, e-learning,vehicular delay-tolerant networks, and mobile and ubiquitous comput-ing. He is the leader of NetGNA Research Group (http://netgna.it.ubi.pt), the Chair of the IEEE ComSoc Technical Committee on eHealth, thePast-chair of the IEEE ComSoc Technical Committee on Communica-tions Software, and Member Representative of the IEEE Communica-tions Society on the IEEE Biometrics Council. He is the editor-in-chiefof the International Journal on E-Health and Medical Communications,the editor-in-chief of the Recent Advances on Communications andNetworking Technology, and editorial board member of several jour-nals. He has been general chair and TPC Chair of many internationalconferences. He is a member of many international TPCs and partici-pated in several international conferences organization. He has authoredor coauthored over 350 papers in refereed international journals andconferences, a book, and 3 patents. He had been awarded the 2014IEEE ComSoc Multimedia Technical Committee Outstanding Leader-ship Award, the Outstanding Leadership Award of IEEE GLOBECOM2010 as CSSMA Symposium Co-Chair and several best papers awards.Prof. Rodrigues is a licensed professional engineer (as senior member),member of the Internet Society, an IARIA fellow, and a senior memberof ACM and IEEE.

Weijian Si is currently a Pro-fessor of Department of Infor-mation & Communication Engi-neering at Harbin EngineeringUniversity, China. He receivedhis Ph.D. degree in College ofInformation and CommunicationEngineering from Harbin Engi-neering University in 2004. Heis the person in charge of wideband signals detection, process-ing and identification in Collegeof Information and Communica-tion Engineering. His researchinterests include wide band sig-

nals detection, high precision passive direction finding and spatial spec-trum estimation.

123

http://netgna.it.ubi.pt

http://netgna.it.ubi.pt

110 L. Wan et al.

Naixing Feng received the B.S.degree in Electronic Science andTechnology and the M.S. degreein Micro- Electronics and Solid-State electronics from TianjinPolytechnic University, Tianjin,China, in 2010 and 2013, respec-tively. He is currently workingtoward his PhD degree in RadioPhysics at Xiamen University,Xiamen. His current researchinterests include computationalelectromagnetics and acoustics.

123

An energy efﬁcient DOA estimation algorithm for ...thealphalab.org/papers/An Energy Efficient DOA Estimation Algorithm... · tuto de Telecomunicações, Por-tugal. He received a

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