Simultaneous Estimation of Azimuth DOA and Angular Spreadof Incident Radio Waves by DOA-Matrix Method Using Planar Array

Makoto Jomoto†, Nobuyoshi Kikuma‡, and Kunio Sakakibara‡†Dept. of Computer Science and Engineering‡Dept. of Electrical and Mechanical Engineering

Nagoya Institute of Technology, Nagoya 466-8555, JapanEmail: kikuma@m.ieice.org

Abstract – In estimating DOA of incident waves with highaccuracy, we often have to take into consideration the angularspread (AS) of each wave due to reflection, diffraction, and scat-tering. As a method of estimating DOA and AS simultaneously,DOA-Matrix method was proposed. In this paper, we extend thearray configuration from the linear array to the planar arrayfor simultaneous estimation of DOA and AS over whole azimuthangles.

Index Terms — DOA, angular spread, DOA-Matrix method,planar array.

1. IntroductionIn order to clarify the radio environments, it is effective

to estimate the DOA of individual incident waves at thereceiving point. For estimating the DOA, the algorithms suchas MUSIC and ESPRIT[1] with array antennas are attractivebecause of high estimation accuracy and high computationalefficiency. However, we often have to take into considerationthe angular spread of each wave due to reflection, diffraction,and scattering[2]. As a method of estimating DOA andAS simultaneously, the use of DOA-Matrix method[3] wasproposed.

In this paper, we extend the array configuration from thelinear array to the planar array for simultaneous estimation ofDOA and AS over whole azimuth angles. Then, we proposeDOA-Matrix method in which the integrated mode vector [3]is applied to the planar array.

2. Array Antenna and Signal ModelFig. 1 shows the K = Kx × Ky element planar array

antenna with element spacing of ∆x and ∆y, which receives Lclustered waves with angular spread in azimuth. We assumethat the Ml element waves of the l-th clustered wave are inphase and continuously distributed in the angular spread ∆θlwith the center angle θl (DOA). Then, the array input vectorx(t) can be expressed as follows.

x(t) =L∑

l=1

sl(t)a(θl,∆θl) + n(t) (1)

a(θl,∆θl) =[a1,1(θl,∆θl), a2,1(θl,∆θl), · · · , aKx,Ky (θl,∆θl)

]T (2)

akx,ky (θ,∆θ) = aokx (θ)aoky (θ)ψkx,ky (θ,∆θ) (3)

aokx (θ) = e− j 2πλ (kx−1)∆x cos θ (kx = 1, · · · ,Kx) (4)

aoky (θ) = e− j 2πλ (ky−1)∆y sin θ (ky = 1, · · · ,Ky) (5)

ψkx,ky (θ,∆θ) =

sinc[π

λ∆θ{(ky − 1)∆y cos θ − (kx − 1)∆x sin θ}

] (6)

where sl(t) is the complex amplitude of the l-th clusteredwave.

3. AS Estimation by DOA-Matrix MethodIf we make the following approximation

ψkx,ky (θ,∆θ) ≃ ψkx,ky+1(θ,∆θ) (7)

then the array input vectors of two subarrays in Fig. 1, x1(t)and x2(t), are expressed as follows.

x1(t) = A1s(t) + n1(t) (8)x2(t) = A2s(t) + n2(t) (9)

A1 = [a1(θ1,∆θ1), · · · , a1(θL,∆θL)] (10)

a1(θl,∆θl) =[a1,1(θl,∆θl), · · · , aKx,Ky−1(θl,∆θl)

]T(11)

A2 = [a2(θ1,∆θ1), · · · , a2(θL,∆θL)] (12)

a2(θl,∆θl) =[a1,2(θl,∆θl), · · · , aKx,Ky (θl,∆θl)

]T(13)

A2 ≃ A1Φ (14)s(t) = [s1(t), s2(t), · · · , sL(t)]T (15)

Φ = diag(ϕ1, · · · , ϕL

)(ϕl = e− j 2π

λ ∆y sin θl ) (16)

where n1(t) and n2(t) are noise vectors of subarray 1 andsubarray 2, respectively.

From the auto-correlation matrix R11 = E[x1(t)xH1 (t)] and

the cross-correlation matrix R21 = E[x2(t)xH1 (t)], we make

R = R21R11−1. Since R has the relation RA1 = A1Φ, we can

obtain the A1 and Φ from the eigendecomposition of R .We derive the DOA and AS estimates over the whole

azimuth angles from A1 and Φ by rearranging the modevector a1 according to the DOA estimates. Specifically, inthe case of 45◦ ≤ | θ | < 135◦, we obtain the AS estimatesfrom comparison between ψkx,ky (θ,∆θ) and ψkx+1,ky (θ,∆θ). Onthe other hand, in the case of 0◦ ≤ | θ | < 45◦ and 135◦ ≤ | θ | <180◦, we obtain the AS estimates from comparison betweenψkx,ky (θ,∆θ) and ψkx,ky+1(θ,∆θ).

4. Computer SimulationComputer simulation is carried out under the conditions

described in Table I. Fig. 2 shows the validity of approxima-tion of (7) or (14) by spatial correlation between a1 and a2.Figs. 3 and 4 show the estimation accuracy as a function ofDOA. In both figures, two methods i.e. the proposed methodand DOA-Matrix method without the rearrangement of modevector are compared.

It is found from Fig. 2 that the value of correlationcoefficient between a1 and a2 is greater than 0.9992 within ASof 10◦. Therefore, it is demonstrated that the approximationof (7) or (14) is valid enough, and also that DOA-Matrix

Proceedings of ISAP2016, Okinawa, Japan

Copyright ©2016 by IEICE

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method is applicable. It is found from Fig. 3 that both methodsprovide high DOA estimation accuracy. Moreover, Fig. 4shows that the proposed method provides more accurate ASestimates with error of 17% or less for all azimuth angles.

5. ConclusionThrough computer simulation, the effectiveness of the pro-

posed method for DOA and AS estimation in whole azimuthangles has been demonstrated. Particularly, it is confirmedthat the AS estimation is improved by rearrangement of themode vector according to the DOA estimates.

In the future work, we will examine the way of removingthe estimation errors due to the overlapping subarrays byusing SLS (Structured Least Squares) algorithm[4].

References

[1] S. U. Pillai, Array signal processing, Springer-Verlag New York Inc.1989.

[2] H. Hotta, N. Kikuma, K. Sakakibara, and H. Hirayama, “Comparisonof MUSIC algorithms using gradient- and integral-type mode vectorsfor direction of arrival and angular spread estimation,” IEICE Trans.Commun. (Japanese Ed.), Vol.J87-B, no.9, pp.1414–1423, Sept. 2004.

[3] M. Okuno, N. Kikuma, H. Hirayama, and K. Sakakibara, “Jointestimation of DOA and angular spread of incident radio waves usingDOA-Matrix method and SAGE algorithm,” IEICE Trans. Commun.(Japanese Ed.), Vol.J98-B, no.2, pp.162–171, Feb. 2015.

[4] M. Jomoto, N. Kikuma, and K. Sakakibara, “Simultaneous estimationof DOA and angular spread of incident radio waves by DOA-Matrixmethod with SLS and SAGE algorithms,” Proc. International Sympo-sium on Antennas & Propagation (ISAP), pp. 344-345, Nov. 2015.

TABLE ISimulation Conditions

Number of elements 49(7 × 7)Element spacing 0.45λ

Number of clustered waves 1DOA of clustered waves −180◦ ∼ 180◦AS of clustered waves 1◦ ∼ 10◦

Number of element wavesin one clustered wave 30

Input SNR 20 [dB]Number of snapshots 30

Number of trials 100

#11#21

#K 1

…

Uniform Linear Array

DirectionOf Arrival

θl

θlm

Angular Spread

Δθl

Ml

Number of element waves

y

Δxx

Δy

#22 #12

…

#1K

#2(K -1)

x

#K Kx

y

…

…

…#K (K -1)

y

x

y

#K 2

y

y

x

Subarray2

Subarray1

#1(K -1)

y#2K…

Fig. 1. Planar rectangular array antenna and incident waveswith angular spread

AS[deg.]

1 2 3 4 5 6 7 8 9 10

Corr

elati

on

bet

wee

n a

an

d a

0.9992

0.9993

0.9994

0.9995

0.9996

0.9997

0.9998

0.9999

1

1.0001

DOA=0[deg.]

DOA=30[deg.]

DOA=60[deg.]

DOA=90[deg.]

12

Fig. 2. Spatial correlation between a1 and a2 vs. AS

DOA [deg.]

-200 -150 -100 -50 0 50 100 150 200

RM

SE

of

DO

A [

deg.]

10-2

10-1

Conventional

Proposed

Fig. 3. RMSE of DOA estimates vs. DOA (AS = 6◦)

DOA [deg.]

-200 -150 -100 -50 0 50 100 150 200

AS

ER

RO

R[%]

0

5

10

15

20

25

30

Conventional

Proposed

Fig. 4. Error of AS estimates vs. DOA (AS = 6◦)

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