Scholarly article on topic 'Frequency and 2D Angle Estimation Based on a Sparse Uniform Array of Electromagnetic Vector Sensors'

Frequency and 2D Angle Estimation Based on a Sparse Uniform Array of Electromagnetic Vector Sensors Academic research paper on "Electrical engineering, electronic engineering, information engineering"

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Academic research paper on topic "Frequency and 2D Angle Estimation Based on a Sparse Uniform Array of Electromagnetic Vector Sensors"

Hindawi Publishing Corporation EURASIP Journal on Applied Signal Processing Volume 2006, Article ID 80720, Pages 1-9 DOI 10.1155/ASP/2006/80720

Frequency and 2D Angle Estimation Based on a Sparse Uniform Array of Electromagnetic Vector Sensors

Fei Ji1 and Sam Kwong2

1 School of Electronic and Information Engineering, South China University of Technology, Guangzhou 510640, China

2 Department of Computer Science, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

Received 25 April 2005; Revised 25 January 2006; Accepted 29 January 2006 Recommended for Publication by Joe C. Chen

We present an ESPRIT-based algorithm that yields extended-aperture two-dimensional (2D) arrival angle and carrier frequency estimates with a sparse uniform array of electromagnetic vector sensors. The ESPRIT-based frequency estimates are first achieved by using the temporal invariance structure out of the two time-delayed sets of data collected from vector sensor array. Each incident source's coarse direction of arrival (DOA) estimation is then obtained through the Poynting vector estimates (using a vector cross-product estimator). The frequency and coarse angle estimate results are used jointly to disambiguate the cyclic phase ambiguities in ESPRIT's eigenvalues when the intervector sensor spacing exceeds a half wavelength. Monte Carlo simulation results verified the effectiveness of the proposed method.

Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

1. INTRODUCTION

The localization of source signals using vector sensor data processing has attracted significant attentions lately. Many advantages of using the vector sensor array have been identified and many array data processing techniques for source localization and polarization estimation using vector sensors have been developed. Nehorai and Paldi developed the Cramer-Rao bound (CRB) and the vector cross-product DOA estimator using the vector cross product of the electric-field and the magnetic-field vector estimates [1,2]. Li [3] developed ESPRIT-based angle and polarization estimation algorithm using an arbitrary array with small loops and short dipoles. Identifiablity and uniqueness study associated with vector sensors were done by Hochwald and Nehorai [4], Ho et al.[5] and Tan et al. [6]. Hochwald and Nehorai [7] studied parameter estimations with application to remote sensing by vector sensors. Ho et al. [8] developed a high-resolution ESPRIT-based method for estimating the DOA of partially polarized sources. Ho et al. [9] further studied the DOA estimation with vector sensors for scenarios where completely and incompletely polarized signals may coexist. Wong [10] has showed that the vector cross-product DOA estimator remains fully applicable for a pair of dipole triad and loop triad spatially displaced by an arbitrary and unknown distance

(rather than being collocated). Uni-vector-sensor ESPRIT is first presented to estimate 2D DOA and the polarization states ofmultiple monochromatic noncoherent incident sources using a single electromagnetic vector sensor by Wong and Zoltowski [11]. Nehorai and Tichavsky [12] presented an adaptive cross-product algorithm for tracking the direction to a moving source using an electromagnetic vector sensor. Ko et al. [13] proposed a structure for adaptively separating, enhancing, and tracking up to three uncorrelated broadband sources with an electromagnetic vector sensor. Wong [14] proposed an ESPRIT-based adaptive geo-location and blind interference rejection scheme for multiple noncooperative wideband fast frequency-hop signals using one electromagnetic vector sensor. The maximum likelihood (ML) and minimum variance distortionless response (MVDR) estimators for signal DOA and polarization parameters for correlated sources are derived by Rahamim et al. [15]. In addition, a novel preprocessing method based on the polarization smoothing algorithm (PSA) for "decorrelating" the signals was also presented. Wong and Zoltowski [16] presented a self-initiating MUSIC-based DOA and polarization estimation algorithm for an arbitrarily spaced array of identically oriented electromagnetic vector sensors. Their proposed algorithm is able to exploit the incident sources' polarization diversity and to decouple the estimation of the sources'

arrival angles from the estimation of the sources' polarization parameters. The same authors further developed a closed-form direction-finding algorithm applicable to multiple arbitrarily spaced vector sensors at possibly unknown locations [17]. A sparse uniform array suffers cyclic ambiguity in its direction-cosine estimates due to the spatial Nyquist sampling theorem. Zoltowski and Wong then further presented another novel ESPRIT-based 2D arrival angle estimation scheme to resolve the aforementioned ambiguity and achieve aperture extension for a sparse uniform array of vector sensors spaced much further apart than a half wavelength [18]. An improved version of the disambiguation algorithm is also presented in [19].

In fact, frequency estimation is a fundamental problem in estimation theory and its applications include radar, array signal processing, and frequency synchronization. For scalar sensor array, a number of ESPRIT-based angle and frequency estimation methods have been proposed. Lemma et al. presented joint angle-frequency estimation method using multidimensional and multiresolution ESPRIT algorithms [20,21]. Zoltowaki and Mathews discuss ESPRIT-based realtime angle-frequency estimation algorithm using scalar sensor array [22].

In this paper, we try to combine the ESPRIT-based frequency estimation with Wong's ESPRIT-based 2D DOA estimation scheme in [18] to yield extended-aperture two-dimensional (2D) arrival angle and carrier frequency estimates with a sparse uniform array of electromagnetic vector sensors. Most of the works mentioned above have previously proposed direction-finding and polarization estimation algorithms using electromagnetic vector sensors; however, this paper is the first in advancing an algorithm for the estimation of both arrival angles and arrival delays.

In the newly proposed algorithm, the ESPRIT-based frequency estimates are achieved using the temporal invariance structure out of two time-delayed sets of data collected from vector sensor array. In that each incident source's direction of arrival (DOA) coarse estimation is obtained through a vector cross-product estimator. Then the frequency estimates and coarse angle estimates results are used jointly to disambiguate the cyclic phase ambiguities in ESPRIT's eigenvalues when the inter vector sensor spacing exceeds a half wavelength.

2. MATHEMATICAL MODEL

Consider the scenario of K uncorrelated monochromatic completely polarized transverse electromagnetic planewaves signals with different carrier frequencies, impinging on an L-shaped array of regularly equally spaced and identical electromagnetic vector sensors from directions (9k, 0 k) and polarization parameters (Yk, nk) (k = 1,...,K). 0 < 9k < n is the kth signal's elevation angle measured from vertical z-axis, 0 < 0k < 2n is azimuth angle, 0 < yk < n/2 is auxiliary polarization angle, and -n < nk < n is polarization phase difference.

The signal source model is given by

where Pk is the kth source's energy, fk is the kth signal's uniformly distributed random phase, and N is the number of independent samples collected by the array. fk is the kth source's digital frequency (between -0.5 and 0.5) normalized to the sampling frequency Fs which satisfies the Nyquist sampling theorem for all the signals' frequencies. Here we normalize to Fs = 1.

A vector sensor contains three electric and three magnetic orthogonal sensors. The spatial response in matrix notation of one vector sensor for the kth signal may be expressed as follows [11]:

sin yk cos 9k cos $kejnk - cos yk sin fk sin yk cos 9k sin fkejnk + cos yk cos fk

def ezk def - sin Yksin 9kejnk

cos yk cos 9k cos fk - sin Yk sin fkejnk - cos yk cos 9k sin fk + sin yk cos fkejnk cos Yk sin 9k

Note that gk does not depend on the signal frequency.

ek =f [exk, eyk, ezk]T and hk d= [hxk, hyk, hzk]T (where the superscript T denotes the vector transpose operator) are orthogonal to each other and the source's direction ofpropaga-tion, that is, the normalized Poynting vector pk [18],

ezk def

Pyk Pzk

ii x ni, i ek|| ||hk|

Uk sin9k cos fk

Vk — sin9k sin fk

_Wk_ cos9k

where * denotes complex conjugation and Uk, Vk, Wk, respectively, symbolize the direction cosine along the x-axis, y-axis, and the z-axis.

The spatial phase factor of the kth signal at the mth vector sensor located (m - 1)A along x-axis equals

qxm (9k, fk) d= ej2nfkFs(m-1^A"k/c

— ej2n/kFs(m-1)A sin 9k cos fk/c

m — 1,2,..., M,

where c is the velocity of light. The spatial phase factor of the kth signal at the Zth vector sensor located (/— 1) A along y-axis equals

qy (9k, fk) — ej2nfkFs(l-1)Avk/c

— ej2nfkFs(l-1)A sin 9k sin fk/c l — 12 L

The 6x1 vector measurement in the nth snapshot is produced by the mth vector sensor along x-axis and the Zth vector sensor along y-axis, respectively,

n(n) —X gkqXm(9k, fk)Sk(n)+ nxm(n),

k—1 K

sk (n) — Tpkej(2nfk"+^k), n — 1,2,..., N,

zy(n) — £ gkqy(9k, fk)Sk(n) + ny(n),

where nxm(n) and ny(n), respectively, symbol 6x1 complex-valued zero-mean additive white noise vector in nth snapshot at the mth vector sensor along f-axis and the Ith vector sensor along y-axis.

Time-delayed data collected from the linear vector sensor array along f-axis is

zxm(n + n0) = X gkq^(0k, $k)Sk(n + n0) + nm (n + n0)

= X gkqm (Ok, <t>k)Sk(n)ej2nfkn0 + nm(n + «o),

are diagonal K X K matrices and are given by

where n0 is the constant sample delay.

We form the following matrices by using (6) and (7):

xi(n) = [zf (n), zf(n),..., zxM_l(n)\ T = AS + N1, yi(n) = [zf(n), zf(n),...,zM(n)]T = AOfS + N2, X2(n) = [z1 (n), zy (n),..., zyL-i(n)]T = BS + N3, y2(n) = [z2 (n), zy (n),..., zL (n)]T = BOy S + N4, y3 (n) = [zf (n + no), zf (n + no),..., zM-1 (n + «o)] T = AOf S + N5,

"W def , N1 = - n1(n) -

_Sk (n)_ _nM-1(n)_

"nf(n)_ def , N3 = - n1(n) - , (9)

_nM (n)_ nL-1(n)

XT def

"n?(n)" def , N5 d=ef n1 (n + no)

_nL (n)_ _nM-1 (n + no)_

A = [af,...,aK] = [qx(0i,00 ® gi>•••>qx(^K,0k) ® gd,

B = [a?,...,aK] = [q?(0f,0i) ® gi,...,q?(9k,0k) ® gK],

qx (9k, 0k ) =

q? (9k, 0k ) =

ej2nfkFsAuk/c

ej2nfkFs(M-2)Auk/c 1

e)2nfkFsAvk/c ej2nfkFs(L-2)Avk/c

A and B are the 6(M - 1) x K and 6(L - 1) x K matrices, respectively, and ® denotes Kronecker product. Of, Oy, and

Ox = diag exp -—:--,... ,exp :

Oy = diag exp -——-,... ,exp :

= diag[exp (j2nf1«o),... ,exp {j2nfKno)].

j 2nfKFsAuK c

j2nfKFsAvK '

X1 X1(1) ' ' X1(N)"

Y1 71(1) ' ' 71(N)_

"X2" "X2(1) ' ' X2(N)"

Y2 _72(1) ' ' 72 (N )

X1 X1(1) ' ' X1 N - no)

Y3 y 3 (1) ' ' 73 (N - no)

From N snapshots, three data sets are formed as the following:

_Y1j L^k1) ' ' ' y 1 (N)_ fel [X2(1) ' ' ' X2(N)] (14)

Z2 LYd Ly2d) '" y2(N)\' (14)

'X. 1 rx, (1) . . . x, (N - n)"

The key problem now is how to estimate the digital frequencies {fk}K=1 and arrival angles {9k,0k}K=1 from the above data sets.

3. ESPRIT-BASED FREQUENCY AND 2D ANGLE ESTIMATION ALGORITHM

From (14), we have formed three distinct matrix-pencil pairs. This first matrix pencil X1 and Y1 has a spatial invariance along the x-axis and can yield estimates of the direction cosines {uk, k = 1,..., K}. This second matrix pencil X2 and Y2 has a spatial invariance along the ?-axis and can yield estimates of the direction cosines {vk, k = 1,..., K}. This third matrix pencil X1 and Y3 has a temporal invariance and can yield estimates of the frequency {fk, k = 1,..., K}.

The first step in ESPRIT is to compute the signalsubspace eigenvectors by eigendecomposing the data correlation matrices R1 = Z1Z1H, R2 = Z2Z2H, and R3 = Z3Z3H (where the superscript H denotes the vector conjugate transpose operator). In the proposed algorithm, we basically modified the algorithm proposed in [18]. Thus, steps 2 to 6 are similar to and taken out from [18].

(!) Deriving the frequency estimates

Let ES denote the 12(M - 1) X K signal-subspace eigenvector matrix whose K columns are the 12(M - 1) X 1 signalsubspace eigenvectors associated with the K largest eigenvalues of R3 = Z3Z3H.

The invariance structure of the matrix-pencil pair implies ES can be decomposed into two 6(M - 1) X K subarrays such that [23]

rEfl " ATf

_E2_ AOf Tf

Because both Ej and E2 are full rank, a unique nonsingular K x K matrix Y* exists such that [11]

e^ = E2

= (Tf )-1

Tf Tf)

can be estimated by the total-least-squares ESPRIT covariance algorithm (TLS-ESPRIT) [23].

^'s eigenvalues equal {[0(]kk = e>2nfk"0, k = 1,...,K},

exp (j2nfkn0) = [&]kk.

If the maximum of the signal digital frequencies is fm n0 is chosen as the following:

I 2nfmaxn0 | < n n0 <

Then we can get the unambiguous frequency estimates:

arg {[^]kkl

where arg{z} is principle argument of the complex number z between -n and n.

(2) Deriving the low-variance but ambiguous estimates of uk

Similarly, for the matrix pencil pair with spatial invariance along the x-axis, Yx's eigenvalues equal {[Ox]kk =

e)2nfkFsAuk/c, k = 1,... , K} .

Because A > Xk (k = 1,...,K) and -1 < Uk <1, there exists a set of cyclically related candidates for the estimation ofuk [18]:

Uk(nu) = ^k +

( - 1 - Vk)

< nu <

fkFsA fkFsA

( - 1 - Vk)

{[®X M

2nfkFs A

where \x] is the smallest integer not less than x; [x\ is the largest integer not greater than x.

(4) Deriving the unambiguous coarse reference estimates of uk and vk from ESPRITs eigenvector

Yx's right eigenvectors constitute the columns of Tx. From [11], we have the following:

A = 0.5{ E1 (Tx)-1 + Ef(Tx )-1 (Ox )-1}. (22)

With noise, the above estimation becomes only approximate.

We have the array manifold estimates from (10):

axk = qx(0k, fk) » gk = qx(dk, fk) »

q! (Ok, fk) êk qXO, fk) hk

qXM-1 (Ok, fk) êk

Âm-1 (Ok, fk) hk.

Define

b;(k) = êX(Ok, fk) êk, 1.

Ci(k) = qX(Ok, fk) hk, (24)

i = 1,2,..., M Note that

bi(k) c*(k)

qXO, fk)ek qf (Ok, fk)h*

|bi(k)H ||ci(k)||

\\qX(Ok, fk) êk| ek h*

M(Ok, fk) hk

So we can get the estimate of Ponyting vector:

— I ■ -1 à

bi(k) c* (k)

|bi(k)| ||ci(k)M"

(3) Deriving the low-variance but ambiguous estimates of vk

Similarly, for the matrix pencil pair with spatial invariance along the y-axis, 's eigenvalues equal {[®y ]kk =

ej2nfkFsAvk/c, k = 1,..., K}.

There exists a set of cyclically related candidates for the estimation of Vk [18]:

Vk (nv) = Vk +

( - 1 - Vk)

< nv <

1 - Vk

2n fkFsA

Unambiguous but high-variance estimates {pxxk, pxyk, pxzk} for {uk, Vk, Wk} have been achieved. This is the so-called vector cross-product estimator who is pioneered by Nehorai and Paldi [1, 2] and firstly adapted to ESPRIT by Wong and Zoltowski [11,24].

Similarly, for the matrix pencil with spatial invariance along the y-axis, we can get another set of unambiguous but high-variance estimates py for {uk, Vk, Wk}. For the matrix pencil with temporal invariance, we can get pk.

(5) Pairing the direction-cosine estimates and frequency estimates

The orderings of {ptxi,ptyi,pfzi, i = 1,2,...,K}, {p*-,pxyj,pj j = 1,2,...,K} and {pyxk,pyyk, pyk, k = 1,2,...,K} are different and need to be paired. {pxi,ptyi, pfzi} can be easily paired

with {pXj,p?j,pZj} and {pXk,P?k,P?k} as follows [18]:

{j'^..., j°} = argmin|

[Px1,-

{ko,...,kK) = argmin||[P1,...,PK] - [pi-

, P jk\

, PL ]

The above minimization is with respect to all possible permutations of {ki,..., kK} and {j1,..., jK}. From pk, pf, py we may form a pk:

ys ~~1

pxk pyk pzk

ysf Av y^V

Pk+Px+Pk

{Pit,pyi,Pi} are already paired with ft, and {pfj,pfyj, pj} with , {pfk,pyyk,pyk} with vk. It follows that {f1,..., fK} is to be paired with {fij0,..., fij«K} and {Vk0,..., v^} [18].

(6) Disambiguation of the low-variance estimates of direction-cosine from ESPRITs eigenvalues [18]

The disambiguated estimates are

uk(nu) = Pk + ■

vk nv = vk +

fkFSA fkFSA

where nu and nv may be separately estimated as

nu = arg min

nv = arg min

pxk - Pk pyk - Vk

nuC fkFS A

nvc fkFSA

(7) The 2D arrival angle estimation

We can calculate low-variance 2D arrival angle estimates from direction-cosine estimates out of ESPRIT's eigenvalues

9k = arcsin (Vk.), vk

0Vk = arctan

Similarly, we can calculate the high-variance 2D arrival angle estimates from direction-cosine estimates out of ES-PRIT's eigenvectors

Vk = arcsin ( ^pXk + p?k),

0k = arctan ( V-^).

Note that pzk maybe applied to judge the quadrant of Ok. 4. SIMULATIONS

Several simulations are presented to verify the effectiveness of the proposed ESPRIT-based frequency and 2D angle estimation algorithm. In these simulations, the total-least-squares

-1o -5 o 5 1o 15 2o 25 3o

SNR (dB)

■■+■■ CRB

—©— From eigenvalue combined with eigenvectors ■ -*- From only eigenvectors

Figure 1: The RMS standard deviation of (9k, 0>k, k = 1,2) versus SNR: the two uncorrelated sources {0j, 02} = (3o°,6o°), {0i, 02} = (4o°, -6o°), {Yi,72} = (o°,45°), {ni, ml = (o°,9o°), {fi, fj} = (o.3,o.4) impinge upon an L-shaped vector sensor, ioo snapshots per experiment, 3oo experiments per data point.

5 io 15 2o 25 3o SNR (dB)

■■*■■ CRB

■ - + ■ ■ Frequency estimates

Figure 2: The RMS standard deviation of (fk, k = 1,2) versus SNR, same settings as an Figure 1.

ESPRIT covariance algorithm (TLS-ESPRIT) [23] is used. We consider the scenario of the two signals impinging one uniform L-shaped array and M = 4, L = 4. All the signal source's energy P is unity and n0 = 1. The intersensor spacing is chosen as A = 10 * Amin/2 (Amin = c/(fmaxFs)) except for the example in Figures 4 and 5.

Figures 1 and 2 give the RMS standard deviations of (Ok, tk, k = 1,2) and (fk, k = 1,2) versus SNR, respectively. The

10-7 -.-.-.-.-.-.-.-

-10 -5 0 5 10 15 20 25 30

SNR (dB)

—s— Angle estimates from only eigenvectors (rad) —*— Frequency estimates ■ ■+■ ■ Angle estimates from eigenvalue combined with eigenvectors (rad)

Figure 3: The RMS bias of 9, (9k, k = 1,2) and (fk, k = 1,2) versus SNR, same settings as in Figure 1.

parameters of the two signals are {9i, 92} = (30°,60°), {(1, (2} = (40°, -60°), {y1, y2} = (0°,45°), {_m, m} = (0°,90°), {f1, f2} = (0.3,0.4). Figure 3 gives the corresponding RMS bias versus SNR. The proposed algorithm successfully resolves all the two electromagnetic source parameters including frequency and 2D angles. Figures 1 and 3 show that the angle estimates from ESPRIT's eigenvalues combined with eigenvectors have better performance than angle estimates obtained from only ESPRIT's eigenvectors at SNR's above 1 dB. It is observed that the RMS bias of angle estimates is less than 0.2° at SNR's above 5dB and 0.1° at SNR's above 10 dB. RMS standard deviation of frequency estimates is less than one order of magnitude greater than the CRB at SNR's above 0 dB.

Figures 4 and 5, respectively, give the RMS standard deviations and bias of (9k, (¡k, k = 1,2) versus intersensor spacing when SNR = 15. The parameters of the two signals are {91,92} = (60°,30°), {(1,(2} = (40°,-60°), {y1, y2} = (0°,45°), {m, = (0° ,90°), {f1, fj = (0.4,0.5). Figure 3 shows that the standard deviations and bias of angle estimates from the eigenvalues combined with eigenvectors decrease as the intersensor spacing increases when A < 60Xmin/2. But the performance of angle estimates obtained from only the eigenvectors remains relatively constant as the inter-sensor spacing increases. Note that when A > 60Xmin/2, the standard deviations and bias of angle estimates from the eigenvalues combined with eigenvectors begin to increase as the intersensor spacing increases. In fact, this phenomenon has been explained in [18].

From (29), it can be seen that the performance of frequency estimation may affect the performance of low-variance angle estimation. Figure 6 gives the RMS standard deviation of (9k, (¡k, k = 1,2) versus SNR. The signal parameters

■■+■■ CRB

-e— From eigenvalue combined with eigenvectors ■ -*- From only eigenvectors

Figure 4: The RMS standard deviation of (6k, (¡k, k = 1,2) versus intersensor spacing when SNR = 15 dB: the two uncorrelated sources {91,92} = (60°,30°), {(1, (2} = (40°, -60°), {y1,y2} = (0°,45°), {n1,n2} = (0°,90°), {f1, f2} = (0.4,0.5) impinge upon an L-shaped vector sensor, 100 snapshots per experiment, 300 experiments per data point.

■ + - ■ From eigenvalue combined with eigenvectors -e— From only eigenvectors

Figure 5: The RMS bias of (9k, (¡k, k = 1,2) versus intersensor spacing when SNR = 15 dB, same settings as in Figure 4.

are the same as in Figure 1 except that {f1, f2} = (0.35,0.4). One curve is calculated from the low-variance angle estimation algorithm when the signal frequencies are not known and estimated. Another curve is calculated by the low-variance angle estimation algorithm when the signal frequencies are known. It is shown that when signal frequencies are known, the RMS standard deviation of angle estimates is

« 10-5 -.-.-.-.-.-.-.-

-10 -5 0 5 10 15 20 25 30

SNR (dB)

■■+■ CRB

—©— Signal frequencies are known ■ -*- Signal frequencies are estimated

Figure 6: The RMS standard deviations of (9k, (¡k, k = 1,2) versus SNR from low-variance angle estimation, same settings as in Figure 1 except that {f1, f2} = (0.35,0.4).

Elevation angle of the first signal (deg)

—e— From eigenvalue combined with eigenvectors ■ -*- From only eigenvectors

Figure 7: The RMS standard deviations of (6k, (¡k, k = 1,2) versus elevation angle of the first signal when SNR = 15 dB. The parameters of the two signals are 92 = 45°, {(1, (2} = (25°, -30°), {y1, y2} = (0°,45°), {m, nil = (0°,90°), {f1, f2} = (0.3,0.4), 100 snapshots per experiment, 300 experiments per data point.

just slightly lower than that when signal frequencies are estimated. Our simulations also show that RMS bias of the low-variance angle estimates when frequencies are known is almost the same as that when frequencies are estimated.

Figure 7 gives the RMS standard deviation of (9k, (¡k, k = 1,2) versus elevation angle of the first signal when SNR = 15 dB. The parameters of the two signals are 92 = 45°, {(1, (2} = (25°, -30°), {y1,y2} = (0°,45°), {n1,n2} = (0°,90°), {f1, f2} = (0.3,0.4). It is observed that the standard deviations of angle estimates from the eigenvalues combined with eigenvectors are greater than angle estimates from ESPRIT eigenvectors when elevation angle nears 90°.

Figure 8 gives the RMS standard deviation of (9k, (¡k, k = 1,2) versus azimuth angle of the first signal when SNR = 15 dB. The signal parameters are the same as in Figure 7 except that {91,92} = (30°,45°), (2 = -30°. It is shown that the RMS standard deviation of angle estimates from two estimation methods almost does not change as the azimuth angle of the first signal is changed.

Figure 9 gives the RMS standard deviation of (9k, (¡k, k = 1,2) and (fk, k = 1,2) versus the number of snapshots when SNR = 15 dB. The parameters of the two signals are {91,92} = (60°,30°), {(1, (2} = (40°, -60°), {y1, y2} = (0°,45°), {n1,m} = (0°,90°), {f1,f} = (0.3,0.4). It is shown that the RMS standard deviation decreases slowly as the number of snapshots increases for the number of snapshots exceeding 50.

Figure 10 gives the RMS standard deviation and bias of (fk, k = 1,2) versus the difference Af of two signal frequencies when SNR = 15 dB. The signal parameters are the same as in Figure 9 except that {f1, f2} = (0.4 - Af,0.4).

=• 10-1 d a

at ima

Azimuth angle of the first signal (deg)

—©— From eigenvalue combined with eigenvectors ■ -*- From only eigenvectors

Figure 8: The RMS standard deviation of (6k, (¡k, k = 1,2) versus azimuth angle of the first signal when SNR = 15 dB, same setting as in Figure 7 except that {91,92} = (30°,45°)and (2 = -30°.

It is observed that when Af is 0.004, the RMS bias is about 2.5e-4 and standard deviation is about 1.6e-3, which shows that two signal frequencies can be separated. Note that just 50 snapshots are used here. For discrete Fourier transform when 50 snapshots are used, the frequency discrimination is just 1/50 = 0.02.

—I— Angle estimates from only eigenvectors (rad) —*— Frequency estimates s— Angle estimates from eigenvalue combined with eigenvectors (rad)

Figure 9: The RMS standard deviation of (Ok, tjk, k = 1,2) and (fk, k = 1,2) versus the number of snapshots when SNR = 15 dB. The parameters of the two signals are {0i, 02} = (60°,30°), t t>2} = (40°, -60°), {y1,y2} = (0°,45°), {^1, mi = (0°,90°), {f1,f2} = (0.3,0.4), 100 snapshots per experiment, 300 experiments per data point.

Difference of two signal frequencies

■ O - RMS standard deviation —I— RMS bias

Figure 10: The RMS standard deviations and bias of (fk, k = 1,2) versus the difference Af of two signal frequencies when SNR = 15 dB, same settings as in Figure 9 except that {f1, f2} = (0.4 -Af ,0.4), 50 snapshots per experiment, 300 experiments per data point.

5. CONCLUSION

In this paper, we propose an ESPRIT-based algorithm that yields 2D angle and frequency estimates. This algorithm can

achieve extended-aperture arrival angle estimation even though using a sparse electromagnetic vector sensor array. Good frequency discrimination obtained even though there are little samples used. Although we only consider the L-shaped array here, the approach may be implemented using a variety of array geometries.

ACKNOWLEDGMENTS

This work is supported by City University of Hong Kong Strategic Grant 7ooi697. This work is done when Fei Ji was visiting City University of Hong Kong.

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Fei Ji received the B.S. degree from the Northwestern Polytechnical University in 1992 and the M.S. and Ph.D. degrees from South China University of Technology in 1995 and 1998. Upon graduation, she joined the Department of Electronic Engineering, South China University of Technology in 1998 as a Lecturer. She worked in the City University of Hong Kong as a Research Assistant from March 2001 to July 2002 and a Senior Research Associate from January 2005 to March 2005. She is currently an Associate Professor in the School of Electronic and Information Engineering, South China University of Technology.

Sam Kwong received his B.S. degree and M.A.S. degree in electrical engineering from the State University of New York at Buffalo, USA and University ofWaterloo, Canada, in 1983 and 1985, respectively. In 1996, he later obtained his Ph.D. degree from the University of Hagen, Germany. From 1985 to 1987, he was a Diagnostic Engineer with the Control Data Canada where he designed the diagnostic software to detect the manufacture faults of the VLSI chips in the Cyber 430 machine. He later joined the Bell Northern Research Canada as a Member of scientific staff. In 1990, he joined the City University of Hong Kong as a Lecturer in the Department of Electronic Engineering. He is currently an Associate Professor in the Department of Computer Science.