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Research ArticleTwo-Dimensional DOA Estimation Using Arbitrary Arrays forMassive MIMO Systems

Alban Doumtsop Lonkeng and Jie Zhuang

School of Communication and Information Engineering, University of Electronic Science and Technology of China (UESTC),Chengdu 611731, China

Correspondence should be addressed to Alban Doumtsop Lonkeng; [email protected]

Received 6 June 2017; Revised 8 August 2017; Accepted 6 September 2017; Published 17 December 2017

Academic Editor: Luciano Tarricone

Copyright © 2017 Alban Doumtsop Lonkeng and Jie Zhuang. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly cited.

With the quick advancement of wireless communication networks, the need for massive multiple-input-multiple-output(MIMO) to offer adequate network capacity has turned out to be apparent. As a portion of array signal processing,direction-of-arrival (DOA) estimation is of indispensable significance to acquire directional data of sources and to empowerthe 3D beamforming. In this paper, the performance of DOA estimation for massive MIMO systems is analyzed andcompared using a low-complexity algorithm. To be exact, the 2D Fourier domain line search (FDLS) MUSIC algorithm isstudied to mutually estimate elevation and azimuth angle, and arbitrary array geometry is utilized to represent massiveMIMO systems. To avoid the computational burden in estimating the data covariance matrix and its eigenvaluedecomposition (EVD) due to the large-scale sensors involved in massive MIMO systems, the reduced-dimension data matrixis applied on the signals received by the array. The performance is examined and contrasted with the 2D MUSIC algorithmfor different types of antenna configuration. Finally, the array resolution is selected to investigate the performance of elevationand azimuth estimation. The effectiveness and advantage of the proposed technique have been proven by detailed simulationsfor different types of MIMO array configuration.

1. Introduction

The use of multiple antennas at the two finishes of wirelesslinks is the consequence of the natural progression of morethan four decades of advancement of adaptive antennatechnology. Key administrations, for example, e-banking,e-learning, and e-well-being, will proceed to prosper andend up being more mobile. On-demand data and entertain-ment will steadily be conveyed over mobile and wirelesscommunication systems. These expansions will prompt anavalanche of mobile and wireless traffic volume, antici-pated to increase a thousandfold throughout the followingdecade [1]. Late advances have shown that multiple-input-multiple-output (MIMO) wireless systems can accomplishgreat increments in the overall system performance. Suchframeworks are expected for the development of newgenerations of mobile radio systems for future wireless

communication standards and applications. Subsequently,it has gotten the consideration not just of the worldwideresearch and development community but also of the wire-less communications industry [2]. There are several signalprocessing functions performed in MIMO systems, amongwhich 3D beamforming for link reliability enhancement hasgotten impressive consideration. To empower the 3D beam-forming, precise estimation of both elevation and azimuthangles of signals is of key significance [3].

Over the last four decades, numerous high-resolutionmethods for estimating the DOA parameters of multiplenarrow-band far-field signal sources, such as multiple signalclassification (MUSIC), estimation of signal parameters viarotational invariance techniques (ESPRIT), weighted sub-space fitting (WSF), and maximum likelihood (ML), havebeen proposed. A large number of these methods are applica-ble only to specific array geometries such as uniform linear

HindawiInternational Journal of Antennas and PropagationVolume 2017, Article ID 6794920, 9 pageshttps://doi.org/10.1155/2017/6794920

https://doi.org/10.1155/2017/6794920

arrays (ULAs), uniform circular arrays (UCAs), or uniformrectangular arrays (URAs). However, in practical cases,the limitations of an array platform usually lead to anunrealistic choice of array geometry from some specificclasses. Furthermore, nonuniform array geometries areempowered to accomplish considerably enhanced resolutionperformance as compared to uniform array geometries with asimilar number of sensors [4].

The automatic weighted subspace fitting (AWSF)algorithm for DOA estimation is demonstrated in [5]. Theestimation accuracy is enhanced, but no implementationhas already been done in MIMO systems to assess theperformance. Algorithms based on sparse representative(SR) procedures [6, 7] deal with the computational complex-ity reduction of the 3D DOA estimation and handle theissues of the assistant calibration for smart transportationframeworks, yet it is substantial for a predefined (URA,ULA, planar array, etc.) array configuration as a vast majorityof DOA algorithms [8, 9]. The DOA estimation task has beenrecast to a probabilistic system in [8, 9], identifying thesmallest angular regions where the approaching signal is inall likelihood. Although effective for some applications, themethod is insufficient for high-resolution analysis since spa-tially close signals cannot be properly sensed. Therefore, amultiresolution technique has been applied to improve thesystem [10]. Notwithstanding the positive and appealinghighlights of previously mentioned approaches, every oneof them shares a similar bottleneck. Undoubtedly, they needthe assessment of the covariance matrix estimated from themeasurements of every sensor at different snapshots. Thisincludes a conspicuous increment in the receiver complexityand a delay in the DOA recovery [11]. In [12], a two-stagefull-dimension DOA estimation scheme based on theMUSIC algorithm is given. It requires an exhaustive mul-tidimensional peak search, resulting in a relatively highcomputational complexity as most 2D DOA estimationalgorithms. The performance of DOA estimation formassive MIMO systems is assessed in [13] using a low-complexity algorithm; good performance was achieved, butthis method is unsubstantial for arbitrary array structure.Therefore, development of low-complexity DOA estimationalgorithms for massive MIMO systems in arbitrary arrayconfiguration becomes extremely necessary for a practicalimplementation perspective.

In this paper, 2D Fourier domain line search MUSIC(FDLSM) is introduced to jointly estimate elevation andazimuth information for massive MIMO systems. Themotivation for applying this algorithm is threefold [14, 15].

(1) It reduces the computational complexity contrastedwith the conventional 2D space search MUSIC orpolynomial rooting strategies, by using 2D fastFourier transform (2D FFT) that avoids the polyno-mial rooting step by replacing it with a computation-ally simple line search procedure.

(2) It provides an improved DOA estimation perfor-mance as compared to the manifold separation tech-nique (MST).

(3) It is formulated in terms of a reduced-dimensiontechnique throughout the whole estimation processof the data covariance matrix, thus decreasing thecomputational burden of estimating the covariancematrix and its eigenvalue decomposition (EVD).

This paper fundamentally concentrates on a low-complexity 2D DOA estimation using the 2D FDLSMalgorithm with a reduced-dimension transformation inmassive MIMO systems. Signal-to-noise ratio (SNR), sourcenumber, and antenna configuration are chosen to assist theevaluation of algorithm performance. Moreover, to the bestof our knowledge, 2D FDLSM is used for the first time toexamine the elevation and azimuth estimation in massiveMIMO systems.

The rest of this paper is sorted out as follows. The sys-tem model and 2D FDLSM are described in Section 2 andSection 3, respectively. The computer simulation and perfor-mance evaluation are presented in Section 4. Section 5concludes the paper.

2. System Model

We consider a MIMO system equipped with M transmitarray sensors and N receive array sensors at the basestation. Both transmit and receive arrays are assumedto be closely located in the arbitrary space so that anytarget located in the far field can be seen at the samedirection by both arrays. All the array sensors are omni-directional. We assume that the number of incidentsources is known and there exist P distinct uncorrelatedsignals coming from directions of (θp, ϕp), p = 1, 2,… , P,where θp and ϕp are the elevation and azimuth angle ofthe pth signal, respectively. The system model is depicted inFigure 1.

(xti, yti, zti)

(xt2, yt2, zt2)

(xri, yri, zri)

(xtM, ytM, ztM)

(xr2, yr2, zr2) (xrN, yrN, zrN)

(xr1, yr1, zr1)(xt1, yt1, zt1)

Incident source

z

𝜆

x

𝜆

y

𝜆

𝜑

𝜃

Figure 1: System model of 2D massive MIMO with an MxNarbitrary array configuration.

2 International Journal of Antennas and Propagation

At the transmit side, M transmit sensors from an Msensor (arbitrary) array and its steering vector at θ, ϕ isgiven by

where λ is the signal wavelength, j = −1 and (xti, yti, zti)are coordinates of the ith transmit sensor, and ⋅ T denotesthe transpose.

For simplicity of notation and without loss of generality,we assume that the steering vector of the receive array is alsoa function of (θ, ϕ), given by

where (xri, yri, zri) are coordinates of the ith receive sensor.The output of the matched filters at the receiver is given

by [16]

x τ =A θ, ϕ s τ + n τ , 3

where s τ = s1 τ , s2 τ ,… , sp τ T ∈ CP×1 is the vector ofsignal waveforms with τ denoting the time interval, n τ isthe B × 1 (B =MN) complex Gaussian white noise vector ofthe zero mean and covariance matrix σ2IB, and A θ, ϕ =a1, a2,… , aP is the B × P steering matrix composed ofP steering vectors with

ap = a θp, ϕp

= ar θp, ϕp ⊗ at θp, ϕp ,4

being the B × 1 steering vector of the pth signal. ⊗ representsthe Kronecker product.

Assume that all impinging signals and noises are uncor-related with each other. Then, the B × B data covariancematrix can be expressed as [16]

Rxx =1L〠L

ℓ=1x τℓ xH τℓ , 5

where L is the snapshot number and ⋅ H stands forthe Hermitian.

3. 2D DOA Estimation Using ArbitraryArrays for Massive MIMO Systems

In this section, a low-DOA estimation algorithm based onthe FDLS MUSIC algorithm to jointly estimate the elevationand azimuth is presented.

3.1. Reduced-Dimension Beamspace (RDBS) Cramer-RaoBound (CRB). The length of the steering vector ap is B,

which is too long and will add high computational burden.Furthermore, it will make the latter estimation of thecovariance matrix in (5) and its EVD costly to implementbecause of the matrix size. So, the reduced-dimensiontransformation is necessary. It has been shown in [15]that there exist a transformation matrix T that makesCRBRDBS T = CRBESP (see (5) for ESP data), which is basedon the following proven theorems [15]:

Theorem 1

E θ̂, ϕ̂ − θ, ϕ θ̂, ϕ̂ − θ, ϕT

≥ CRBESP, 6

where

CRBESP =σ2

2L Re WHP⊥AW ⊙ SAHR−1

xxAST −1

7

Here, θ̂, ϕ̂ , θ, ϕ , S, and ⊙ denote the estimated DOAs,the true DOAs, the signal covariance matrix, and the Schurproduct, respectively.

P⊥A = IB −A AHA −1AH is the orthogonal projection onto

the null space of AH and

W = w θ1, ϕ1 ,… ,w θP, ϕP ,

w θp, ϕp =∂a θp, ϕp∂ θp, ϕp

,

p = 1,… , P

8

The expression of CRBRDBS T is obtained by substitutionof A, W and Rxx with THA, THW, and THRxxT, respectively.

Theorem 2. Assume that k ≥ 2P and that T is the B × kmatrixfulfilling THT = Ik. Then, if

Ψ Α θ, ϕ W θ, ϕ ⊆Ψ T , 9

the relation CRBRDBS T, θ, ϕ = CRBESP θ, ϕ holds true.

at θp, ϕp = ej2πλ

xt1sin θpcosϕp+yt1 sinθp sinϕp+zt1cosθp ,… , ej2πλ xtMsinθp cosϕp+ytM sinθp sinϕp+ztM cosθpT, 1

ar θp, ϕp = ej2πλ

xr1sinθp cosϕp+yr1sinθpsinϕp+zr1cosθp ,… , ej2πλ xrN sinθp cosϕp+yrN sinθpsinϕp+zrN cosθpT, 2

3International Journal of Antennas and Propagation

Here,Ψ ⋅ denotes the range space of the matrix and ⊆ isread as “is contained in.”

The proofs of the above theorems can be found in [15] andthe references therein. After various simulations, k = 4P wasfound to be the minimum value meeting the requirement interms of accuracy.

According to [15], the matrix T ∈ℂB×k can be obtainedby applying the singular value decomposition (SVD) ofA W . Let

A W =UΣVH 10

be the SVD of A W for the k largest singular values, whereU is the B × k, Σ = diag ς1,… , ςk , and ςk ≥ ςk ≥⋯≥ ςkand V is the k × k. Since U spans the range space of A W ,one may take T =U.

Include the B × k matrix T, P < k ≤ B, and the map-ping x↦y = THx from ESP to RDBS, a new set of observa-tion result, consisting of k-dimensional vectors. Then, inRDBS, the representation of the signals received by the arraywill be

y τ = THx τ

= THA θ, ϕ s τ + THn τ11

For the signal model in (11), the covariance matrix Ryycan be estimated with L snapshots by

Ryy =1L〠L

ℓ=1y τℓ yH τℓ 12

The eigen decomposition of the sample covariance matrix(12) yields

Ryy = EsΛsEHs + EnΛnEH

n , 13

where the sample eigenvalues are again sorted in descendingorder (λ1 ≥ λ2 ≥⋯ ≥ λk); the matrices Es ≅ e1, e2,… , ePand En ≅ eP+1, λP+2,… , ek contain in their columns thesignal and noise subspace eigenvectors of (13), respectively.Correspondingly, the diagonal matrices Λs ≅ diag λ1, λ2,… ,λP andΛn ≅ diag λP+1, λP+2,… , λk are built from the signaland noise subspace eigenvalues of (13), respectively.

3.2. 2D FDLS MUSIC-Based DOA Estimation. This sectionextends the idea of [14] into the 2D DFT to solve efficiently(14). The procedure is as follows:

By combining (4) and (13) and using the transforma-tion matrix T, the MUSIC null spectrum in [17] can berewritten as

f θ, ϕ = THa θ, ϕ HEnEHn THa θ, ϕ

= aH θ, ϕ T EnEHn THa θ, ϕ

= ∥EHn T

Ha θ, ϕ ∥2,14

where ∥⋅ ∥ is the vector 2-norm.From (14), it can be seen that if θ, ϕ is the desired DOA,

f θ, ϕ will be equivalent to zero. Given that f θ, ϕ is a

periodic function in θ and ϕ with the period 2π, it can beexpressed using the finite 2D Fourier series as [18]

f θ, ϕ ≈ 〠Qe−1

m=− Qe−1〠Qa−1

n=− Qa−1Gmne

jmθejnϕ, 15

where the Fourier coefficients are given by

Gmn ≈14π2 〠

Qe−1

qe=− Qe−1〠Qa−1

qa=− Qa−1f k1, k2 e −jmk1 e −jnk2 , 16

with k1 = qeΔθ, k2 = qaΔϕ, Δθ = 2π/ 2Qe − 1 , and Δϕ = 2π/2Qa − 1 , where 2Qe − 1 and 2Qa − 1 represent the sam-ple number in elevation and azimuth, respectively.

We notice that 2D FFT can be used to obtain the Fouriercoefficient matrix G ∈ C 2Qe−1 × 2Qa−1 rapidly. Once the Fou-rier coefficients are acquired, to ameliorate the resolution,zero padding is applied efficiently as follows:

Gmn =Gmn, if ∣m∣ < 2Qe − 1 and n∣ < 2Qa − 1,0, if 2Qe − 1 < ∣m∣ < J0 and 2Qa − 1 < ∣n∣ < J1

17

Generally, we pick J0 ≫ 2Qe − 1 and J1 ≫ 2Qa − 1. Thenew cost function f θ, ϕ of every grid can be computedeffectively by applying 2D IFFT to the zero padded Fouriercoefficient matrixG ∈ CJ0×J1 in (17). Accordingly, no polyno-mial rooting is required any longer as the polynomial rootingstep is replaced by a simple line search with θ and ϕ.

Recall that 1/f θ, ϕ will assume a very large valuewhen θ, ϕ is equal to the DOA of one of the signals. Thus,the P signals can be easily obtained by taking the P greatestpeaks of 1/f θ, ϕ .

We also recall that the computational complexities of theconventional 2D space search MUSIC algorithm and the2D FFT-based technique are entirely unequal. For the con-ventional 2D space search MUSIC algorithm, to obtainsuitable accuracy, the intrigued area has to be divided intoexceptionally dense grids. For instance, if the intriguedarea is θ ∈ 0∘, 45∘ and ϕ ∈ −50∘, 50∘ with 0 1∘ × 0 1∘ asthe unit grid, we have to compute 451 × 1001 = 451451 timesthe matrix-vector product ∥EH

n THa θ, ϕ ∥2 in (14). However,in the 2D FFT technique, only 2Qe − 1 × 2Qa − 1 matrix-vector products in (14) need to be computed (Qe =Qa = 77was used for our test). For other grids, zero padding isapplied straightforwardly; hence, the computational com-plexity is decreased significantly. Furthermore, this techniquehas a lower complexity than the 2D polynomial rootingstrategy. For instance, to estimate the 2D DOAs in [19],additional time is required to solve the high-order bivariatenonlinear polynomials. Since rooting a bivariate nonlinearpolynomial is costly and the 2D FFT/IFFT method forreal-time implementation is promptly accessible, the 2DFFT-based approach comes out to be competitive.


3.3. Summary. The major steps of the proposed algorithm for2D DOA estimation in arbitrary geometry for the MIMOsystem are as follows:

(1) Obtain the RDBS matrix T from the SVD of (10) (fork = 4P); then, apply the mapping x↦y = THx toacquire the new representation of the array observa-tions (11).

(2) From (11), construct the covariance matrix Ryy in(12) and perform the eigenvalue decomposition toobtain the noise subspace En.

(3) Use 2Qe − 1 × 2Qa − 1 samples to compute thecost function in (14).

(4) Compute the Fourier coefficient matrix G ∈C 2Qe−1 × 2Qa−1 by applying the 2D FFT.

(5) Employ the zero padding technique described in (17)to shape the extensive matrix G ∈ CJ0×J1 .

(6) Apply the 2D IFFT to G to obtain the new costfunction f θ, ϕ .

(7) Estimate the desired DOAs from the null spectrum1/f θ, ϕ by taking the P largest peaks.

3.4. Computational Complexity. The order of computationalcomplexity of the proposed FD line search MUSIC forMIMO systems with a reduced-dimension matrix is com-pared with that of the case where the reduced-dimensionmatrix is not used. The results are shown in Table 1.

The last row of Table 1 shows the total number of opera-tions (addition andmultiplication) needed for both cases (i.e.,B = 256, P = 5, L = 100, and J0 = J1 = 2000). We can easily seethat, despite the fact that our approach uses the SVD, thecomputational complexity is reduced by 53% as comparedwith the case where the mapping matrix is not used.

Table 2 gives the elapsed time by the two methods, fromwhich it can be easily seen that the enhancement of the speedof our method is considerable.

4. Simulation Results

To get a more quantitative understanding of how the 2DFourier domain (FD) line search MUSIC algorithm performsin massive MIMO systems, Monte Carlo simulations areconducted in this section. In the simulation, the totalnumber of antenna elements is set to 256, which is consid-ered one possible dimension of massive MIMO [1]. Thecase where there are 5 sources is used with ϕ = −70∘, −40∘,−13∘, 60∘, 80∘ and θ = 40∘, 31∘, 60∘, 70∘, 50∘ . The simulationparameters are listed in Table 3.

Figure 2 shows the 2D spectrum of the null-spectrumfunction of arbitrary geometry (SNR=10dB) for differentMIMO configurations with 256 antennas 16 × 16, 64 × 4,4 × 64, 32 × 8, and 8 × 32 .

4.1. Performance Evaluation. The root-mean-squared error(RMSE) has been used to measure the effectiveness of ourproposed 2D DOA estimation algorithm, which is definedas [13]

RMSE = 1P

1Nc

〠P

p=1〠Nc

n=1θ̂p,n − θp

2+ ϕ̂p,n − ϕp

2, 18

where θ̂p,n is the estimated elevation angle θp at the nthMonte Carlo trial, which is similar for ϕ̂p,n. P is the estimatedsource number and Nc represents the Monte Carlo simula-tion times. The results are shown in Figure 3.

From the RMSE results for massive MIMO DOAestimation that are illustrated in Figure 3, we can see thatthe RMSE values of our proposed 2D FDLS MUSIC are

Table 1: Computational complexity.

Algorithm Using reduced-dimension matrix Without using reduced-dimension matrix

SVD O min B2k, Bk2 —

Covariance matrix O k2L + P3 O B2L + P3

EVD O k3 O B3

2D FDLS MUSIC O QeQakL + J0 J1 log 2 J0 J1 O QeQa B L + J0 J1 log 2 J0 J1

Total O 72, 815, 745 O 235, 920, 561

Table 2: Elapsed time of the two methods (100-simulation time,AMD E-450 APU 2∗1.65GHz, Windows 7 pro 64 bit, 2 GBmemory, Matlab 8.4).

Method Total time (seconds)

Using reduced-dimension matrix 49 471815Without using reduced-dimension matrix 104 942036

Table 3: Simulation parameters.

Parameters Setting

Antenna configuration Arbitrary array

Antenna elements 256Center frequency 3 0GhzSnapshots 100Number of Monte Carlo 100Directivity of the base station antennas Omnidirectional


below 0 065 with SNR ranging from 0dB to 30 dB. Further-more, the DOA can be clearly observed; therefore, we canconclude that the proposed algorithm works well in massiveMIMO systems. Different curves correspond to differenttypes of antenna configuration as indicated in the legend.As the SNR increases, the performance of each array config-uration becomes much more better. It is interesting to seethat the 64 × 4 and 4 × 64 arrays have produced excellentquality compared with other array configurations.

Figure 2 also compares our proposed algorithm with the2D MUSIC (using uniform circular array (UCA) structure).It shows that for SNR ranging from 10 to 30dB, both algo-rithms provide almost the same accuracy (for 64 × 4, 4 × 64,8 × 32, and 32 × 8 configurations) while our approach usesless complexity as explained in Section 3.2.

From Figure 4, we can clearly observe that for the numberof sensors varying from 50 to 256 elements, our approach stilloffers better performance than the traditional 2D MUSIC(UCA and SNR=20dB were used).

4.2. Performance Comparison between Elevation andAzimuth Estimation. Figure 3 shows that two identicalantenna configurations can have different performance. Tobetter understand it, we have plotted the RMSE comparisonbetween elevation and azimuth estimation of five sourceswith four types of array configuration (notice that 16 × 16array configuration is not included due to its higher RMSEvalues); the results are shown in Figure 5.

It is easy to see from this figure that the overall perfor-mance of angle estimations weighs more in some cases while

2D S

pect

rum

(dB)

2D FDLS MUSIC for MIMO

−20

−25

−30

−500

20 40 60 80500Azimuth (degree) Elevation (degree)

(a) 64 × 4 configuration


2D S

pect

rum

(dB)

Azimuth (degree) Elevation (degree)

−20

−25

−30

−500 20 40 60 8050

0

(b) 4 × 64 configuration


−20

−25

−30

−35

−500

0 20 40 608050

2D S

pect

rum

(dB)


(c) 32 × 8 configuration


2D S

pect

rum

(dB)

Azimuth (degree) Elevation (degree)−500 20 40

60 800

50

−35

−30

−25

−20

(d) 8 × 32 configuration


2D S

pect

rum

(dB)


−20

−25

−30

−35

−500 20 40 60 80

050

(e) 16 × 16 configuration

Figure 2: 2D FDLS MUSIC spectrum for MIMO in different antenna configurations.


the accuracy of azimuth/elevation angles is more valued inother cases. Thus, the antenna array configuration can beadjusted as an efficient way to better satisfy the requirementsof DOA estimation. Based on the simulations above, the

64 × 4 and 4 × 64 have the best performance of the overallDOA estimation. It is interesting to see that in the aspectof azimuth estimation, the 64 × 4 and 4 × 64 perform wellwhile the 32 × 8 and 8 × 32 have good performance inelevation estimation. Figure 6 presents angle (azimuth andelevation) estimation results of our algorithm for all fivessources with SNR=10dB using 64 × 4 configuration.

4.3. Performance Comparison between Array Configurationand Source Number. Figure 7 depicts the RMSE versus sourcenumber for SNR=10dB with different types of sensor con-figuration. It shows that the RMSE increases gradually alongwith the growth of sources. Particularly, the 64 × 4 and 4 × 64have the best performance.

4.4. Performance Comparison between Array Configurationand Snapshot Number. The RMSE for massive MIMODOA estimation versus snapshot number for SNR=10dBwith different types of array configuration is plotted inFigure 8. It shows that the DOA estimation performanceis getting better with the increase in the number of snap-shots. That is because the error in estimating the signalcovariance matrix in (12) is reduced with the increase inthe snapshot number.

5. Conclusion

In this paper, a 2D DOA estimator in arbitrary arraygeometry for massive MIMO has been analyzed and mod-eled based on low-complexity 2D Fourier domain line

SNR (dB)0 5 10 15 20 25 30

RMSE

(deg

rees

)

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

2D FDLS with array configuration (16 × 16)2D MUSIC with array configuration (16 × 16)2D FDLS with array configuration (32 × 8)2D MUSIC with array configuration (32 × 8)2D FDLS with array configuration (8 × 32)2D MUSIC with array configuration (8 × 32)2D FDLS with array configuration (64 × 4)2D MUSIC with array configuration (64 × 4)2D FDLS with array configuration (4 × 64)2D MUSIC with array configuration (4 × 64)

Figure 3: RMSE comparison between 2D MUSIC and 2D FDLSMUSIC algorithms for massive MIMO DOA estimation versusSNR for five sources with different types of array configuration.

Number of sensors50 100 150 200 250 300

RMSE

(deg

rees

)

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

2D MUSIC2D FDLS MUSIC

Figure 4: RMSE comparison between 2D MUSIC and 2D FDLSversus sensor number.

SNR (dB)0 5 10 15 20 25 30

RMSE

(deg

rees

)

0.022

0.024

0.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04

Elevation with array configuration (8 × 32)Azimuth with array configuration (8 × 32)

Azimuth with array configuration (32 × 8)Elevation with array configuration (32 × 8)Azimuth with array configuration (64 × 4)Elevation with array configuration (64 × 4)Azimuth with array configuration (4 × 64)Elevation with array configuration (4 × 64)

Figure 5: RMSE comparison between elevation and azimuth of fivesources with eight types of array configuration.


search MUSIC algorithm. The algorithm utilizes the 2DFFT to compute the null-spectrum function. It does notrequire a polynomial root-finding procedure to jointly esti-mate elevation and azimuth information, and it performsin a high quality in massive MIMO as shown in simula-tion results. Moreover, to avoid a large size of the covari-ance matrix and further decrease the computationalcomplexity of its EVD, the reduced-dimension transforma-tion has been applied on the received signals. The effec-tiveness and advantage of the proposed technique havebeen shown by detailed simulations for different types ofMIMO array configuration.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by the National Natural Sci-ence Foundation of China (NSFC) under Grant 61571090and the Fundamental Research Funds for the Central Uni-versities under Grant ZYGX2014J007.

Test number0 5 10 15 20 25 30 35 40 45 50

Azi

mut

h (d

egre

es)

−80

−60

−40

−20

0

20

40

60

80

(a) Estimation results of azimuth angle

Test number0 5 10 15 20 25 30 35 40 45 50

Elev

atio

n (d

egre

es)

30

35

40

45

50

55

60

65

70

75

(b) Estimation results of elevation angle

Figure 6: Estimation results of DOA with 10 dB using 64 × 4 configuration.

Source number1 1.5 2 2.5 3 3.5 4 4.5 5

RMSE

(deg

rees

)

0.04

0.045

0.05

0.055

0.06

0.065

0.07

2D FDLS with array configuration (16 × 16)2D FDLS with array configuration (8 × 32)2D FDLS with array configuration (32 × 8)2D FDLS with array configuration (4 × 64)2D FDLS with array configuration (64 × 4)

Figure 7: RMSE for massive MIMO DOA estimation versus sourcenumber for SNR= 10 dB with different types of array configuration.

Snapshot number50 100 150 200 250 300

RMSE

(deg

rees

)

0.042

0.044

0.046

0.048

0.05

0.052

0.054

0.056

0.058

0.06

0.062

2D FDLS with array configuration (16 × 16)2D FDLS with array configuration (8 × 32)2D FDLS with array configuration (32 × 8)2D FDLS with array configuration (4 × 64)2D FDLS with array configuration (64 × 4)

Figure 8: RMSE for massive MIMO DOA estimation versussnapshot number for SNR= 10 dB with different types of arrayconfiguration.


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