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
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
Fei Ji1 and Sam Kwong2
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
(rather than being collocated). Uni-vector-sensor ESPRIT is first presented to estimate 2D DOA and the polariza- tion states of multiple 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 electromag- netic vector sensor. The maximum likelihood (ML) and min- imum 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 estima- tion algorithm for an arbitrarily spaced array of identically oriented electromagnetic vector sensors. Their proposed al- gorithm is able to exploit the incident sources’ polariza- tion diversity and to decouple the estimation of the sources’ 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 identi- fied and many array data processing techniques for source localization and polarization estimation using vector sen- sors have been developed. Nehorai and Paldi developed the Cram´er-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] de- veloped ESPRIT-based angle and polarization estimation al- gorithm 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] stud- ied 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 es- timation with vector sensors for scenarios where completely and incompletely polarized signals may coexist. Wong [10] has showed that the vector cross-product DOA estimator re- mains fully applicable for a pair of dipole triad and loop triad spatially displaced by an arbitrary and unknown distance
2 EURASIP Journal on Applied Signal Processing
⎡
⎤
⎡
where Pk is the kth source’s energy, ϕk 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) normal- ized to the sampling frequency Fs which satisfies the Nyquist sampling theorem for all the signals’ frequencies. Here we normalize to Fs =1.
def=
def=
A vector sensor contains three electric and three mag- netic orthogonal sensors. The spatial response in matrix no- tation of one vector sensor for the kth signal may be ex- pressed as follows [11]: ⎤ arrival angles from the estimation of the sources’ polarization parameters. The same authors further developed a closed- form direction-finding algorithm applicable to multiple ar- bitrarily spaced vector sensors at possibly unknown loca- tions [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 pre- sented another novel ESPRIT-based 2D arrival angle estima- tion scheme to resolve the aforementioned ambiguity and achieve aperture extension for a sparse uniform array of vec- tor sensors spaced much further apart than a half wavelength [18]. An improved version of the disambiguation algorithm is also presented in [19].
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
. gk
exk eyk ezk hxk hyk hzk sin γk cos θk cos φke jηk − cos γk sin φk sin γk cos θk sin φke jηk + cos γk cos φk − sin γk sin θke jηk − cos γk cos θk cos φk − sin γk sin φke jηk − cos γk cos θk sin φk + sin γk cos φke jηk cos γk sin θk (2)
In fact, frequency estimation is a fundamental problem in estimation theory and its applications include radar, ar- ray signal processing, and frequency synchronization. For scalar sensor array, a number of ESPRIT-based angle and fre- quency 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 real- time angle-frequency estimation algorithm using scalar sen- sor array [22].
⎡
⎤
⎤
⎡
⎡
(cid:9)
⎢ ⎣
⎥ ⎦ =
⎢ ⎣
⎢ ⎣
Note that gk does not depend on the signal frequency. def= [hxk, hyk, hzk]T (where the su- def= [exk, eyk, ezk]T and hk ek perscript T denotes the vector transpose operator) are or- thogonal to each other and the source’s direction of propaga- tion, that is, the normalized Poynting vector pk [18],
⎥ ⎦ ,
(cid:9) (cid:9) =
⎤ ⎥ ⎦ = ek(cid:9) (cid:9)ek
(cid:9) × h∗ k(cid:9) (cid:9)hk
pk = pxk pyk pzk uk vk wk sinθk cos φk sinθk sin φk cosθk (3)
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 esti- mates with a sparse uniform array of electromagnetic vector sensors. Most of the works mentioned above have previously proposed direction-finding and polarization estimation al- gorithms using electromagnetic vector sensors; however, this paper is the first in advancing an algorithm for the estimation of both arrival angles and arrival delays.
(cid:10)
(cid:11)
where ∗ denotes complex conjugation and uk, vk, wk, respec- tively, 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)Δ along x-axis equals
def= e j2π fkFs(m−1)Δuk/c = e j2π fkFs(m−1)Δ sin θk cos φk/c, m = 1, 2, . . . , M,
qx m θk, φk
(4) In the newly proposed algorithm, the ESPRIT-based fre- quency 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 intervector sensor spacing exceeds a half wavelength.
2. MATHEMATICAL MODEL
(cid:11)
= e j2π fkFs(l−1)Δvk/c = e j2π fkFs(l−1)Δ sin θk sin φk/c,
where c is the velocity of light. The spatial phase factor of the kth signal at the lth vector sensor located (l−1)Δ along y-axis equals (cid:10) θk, φk q y l (5) l = 1, 2, . . . , L.
K(cid:12)
The 6×1 vector measurement in the nth snapshot is pro- duced by the mth vector sensor along x-axis and the lth vec- tor sensor along y-axis, respectively,
(cid:10) θk, φk
(cid:11) Sk(n) + nx
m(n),
k=1 K(cid:12)
(cid:11)
(cid:2)
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 elec- tromagnetic vector sensors from directions (θk, φk) and po- larization parameters (γk, ηk) (k = 1, . . . , K). 0 ≤ θk < π is the kth signal’s elevation angle measured from vertical z- axis, 0 ≤ φk < 2π is azimuth angle, 0 ≤ γk < π/2 is auxiliary polarization angle, and −π ≤ ηk < π is polarization phase difference. zx m(n) = gkqx m The signal source model is given by (6)
(cid:10) θk, φk
gkq y Sk(n) + n
y l (n) = z
l
y l (n),
k=1
sk(n) = Pke j(2π fkn+ϕk), n = 1, 2, . . . , N, (1)
F. Ji and S. Kwong 3
m(n) and n
(cid:15)
Φt
(cid:16) ,
(cid:15)
(cid:16)
y where nx l (n), respectively, symbol 6×1 complex- valued zero-mean additive white noise vector in nth snapshot at the mth vector sensor along x-axis and the lth vector sensor along y-axis.
exp Φx = diag , . . . , exp
(cid:11)(cid:14)
(cid:10)
(cid:13)
(cid:11) , . . . , exp
K(cid:12)
(cid:11)
(cid:11)
(cid:11)
=
(cid:11) Sk
Time-delayed data collected from the linear vector sensor exp , array along x-axis is , . . . , exp (cid:10) . Φy = diag Φt = diag exp are diagonal K × K matrices and are given by j2π f1FSΔu1 c j2π f1FSΔv1 c j2π f1n0 j2π fK FSΔuK c j2π fK FSΔvK c j2π fK n0
(cid:10) θk, φk
(cid:10) n + n0
(cid:10) n + n0
(cid:10) n + n0
k=1 K(cid:12)
zx m gkqx m + nx m
(cid:17)
(cid:18)
(cid:17)
(cid:18)
=
(cid:10) θk, φk
(cid:11) Sk(n)e j2π fkn0 + nx m
(cid:11) ,
(cid:10) n + n0
k=1
=
(cid:17)
(cid:18)
(cid:18)
(cid:17)
=
(13) From N snapshots, three data sets are formed as the follow- ing: gkqx m , Z1 = X1 Y1 (7)
(cid:17)
(cid:14)T
=
= AS + N1,
where n0 is the constant sample delay. (14) We form the following matrices by using (6) and (7): Z2 = (cid:17) X2 Y2 (cid:18) , (cid:18) (cid:11) (cid:11) . Z3 = x1(n) = X1 Y3 x1(1) · · · x1(N) y1(1) · · · y1(N) x2(1) · · · x2(N) y2(1) · · · y2(N) (cid:10) N − n0 (cid:10) N − n0 x1(1) · · · x1 y3(1) · · · y3
= AΦx (cid:14)T
= BS + N3,
(cid:11)(cid:14)T
M−1(n) (cid:14)T M(n) y L−1(n) (cid:14)T y L(n) (cid:11) , . . . , zx
y1(n) = S + N2, The key problem now is how to estimate the digital fre- k=1 and arrival angles {θk, φk}K k=1 from the x2(n) = quencies { fk}K above data sets. (8) y2(n) = 3. ESPRIT-BASED FREQUENCY AND 2D ANGLE y3(n) =
= BΦy S + N4, (cid:10) n + n0
M−1
2(n), . . . , zx 3(n), . . . , zx y 2 (n), . . . , z y 3 (n), . . . , z (cid:10) (cid:11) , zx n + n0 2
ESTIMATION ALGORITHM
(cid:13) 1(n), zx zx (cid:13) 2(n), zx zx (cid:13) y 1 (n), z z (cid:13) y 2 (n), z z (cid:13) (cid:10) zx n + n0 1 = AΦt
S + N5,
⎡
⎡
⎤
⎤
def=
where
⎢ ⎢ ⎣
⎢ ⎢ ⎣
⎥ ⎥ ⎦ ,
⎥ ⎥ ⎦ ,
⎤
⎤
S def= N1
def=
def=
⎢ ⎢ ⎣
⎢ ⎢ ⎣
⎥ ⎥ ⎦ ,
⎥ ⎥ ⎦ ,
From (14), we have formed three distinct matrix-pencil pairs. This first matrix pencil X1 and Y1 has a spatial invari- ance 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 y-axis and can yield es- timates 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 }.
H
H
N2 N3 (9)
(cid:11)
⎡
⎡
⎤
def=
def=
⎢ ⎢ ⎣
⎢ ⎢ ⎣
⎥ ⎥ ⎦ ,
, and R3 = Z3Z3 , R2 = Z2Z2
(cid:11)
N4 N5
(cid:11)
(cid:11)
(cid:13)
=
K
The first step in ESPRIT is to compute the signal- subspace eigenvectors by eigendecomposing the data corre- H lation matrices R1 = Z1Z1 (where the superscript H denotes the vector conjugate trans- pose operator). In the proposed algorithm, we basically mod- ified the algorithm proposed in [18]. Thus, steps 2 to 6 are similar to and taken out from [18].
⊗ gK
(cid:10) θ1, φ1
(cid:14)
(cid:11)
(cid:11)
=
A = qx nx 1(n) ... nx M−1(n) ⎡ y 1 (n) n ... y L−1(n) n (cid:10) nx n + n0 1 ... (cid:10) n + n0 (cid:10) θK , φK nx M−1 ⊗ g1, . . . , qx S1(n) ... SK (n) ⎡ nx 2(n) ... nx M(n) ⎤ y 2 (n) n ⎥ ... ⎥ ⎦ , y L(n) n (cid:13) (cid:14) 1, . . . , ax ax (1) Deriving the frequency estimates
(cid:13) qy
(cid:10) θK , φK
⊗ gK
(cid:10) θ1, φ1
⊗ g1, . . . , qy
B =
(cid:13) y y 1 , . . . , a a K
(cid:14) , (10) (cid:14) , (11)
⎡
⎤
H
(cid:11)
def=
. Let Et S denote the 12(M − 1) × K signal-subspace eigenvec- tor matrix whose K columns are the 12(M − 1) × 1 signal- subspace eigenvectors associated with the K largest eigenval- ues of R3 = Z3Z3
(cid:10) θk, φk
⎢ ⎢ ⎢ ⎢ ⎣
⎥ ⎥ ⎥ ⎥ ⎦
(cid:18)
(cid:17)
(cid:18)
(cid:17)
⎤
=
qx , The invariance structure of the matrix-pencil pair im- S can be decomposed into two 6(M − 1) × K subarrays plies Et such that [23] 1 e j2π fkFSΔuk/c ... e j2π fkFS(M−2)Δuk/c ⎡ (12) . (15) Et S = AT AΦt
t Tt
(cid:11)
def=
(cid:10) θk, φk
⎥ ⎥ ⎥ ⎥ ⎦
⎢ ⎢ ⎢ ⎢ ⎣
2 are full rank, a unique nonsingu-
. qy Because both Et Et 1 Et 2 1 and Et 1 e j2π fkFSΔvk/c ... e j2π fkFS(L−2)Δvk/c lar K × K matrix Ψt exists such that [11]
=⇒ ATtΨt = AΦt (cid:11)−1Φt
(cid:11)−1.
Tt Et 1 (16) Ψt = Et 2 =⇒ Ψt = A and B are the 6(M − 1) × K and 6(L − 1) × K matrices, , and respectively, and ⊗ denotes Kronecker product. Φx , Φy
(cid:10) Tt
Tt =⇒ Φt = TtΨt(cid:10) Tt
4 EURASIP Journal on Applied Signal Processing
(cid:10)
(cid:13)
=
(cid:14) kk.
(cid:11)−1(cid:10)
Ψt (4) Deriving the unambiguous coarse reference can be estimated by the total-least-squares ESPRIT estimates of uk and vk from ESPRIT’s eigenvector covariance algorithm (TLS-ESPRIT) [23]. Ψt ’s right eigenvectors constitute the columns of Tx. From ’s eigenvalues equal {[Φt]kk = e j2π fkn0 , k = 1, . . . , K }, (cid:11) Ψx [11], we have the following: Φt exp (17) j2π (cid:19)fkn0
(cid:22) .
(cid:21) (cid:19)A = 0.5
Φx(cid:11)−1
(cid:10) Tx
(cid:11)−1 + Ex
(cid:10) Tx
2
(cid:20) (cid:20)2π fmaxn0
(cid:20) (cid:20) ≤ π =⇒ n0 ≤
(cid:20) (cid:20) .
(cid:18)
(cid:17)
(cid:11)
(cid:11)
⊗
(cid:10) θk, φk
(cid:22)
(22) Ex 1 If the maximum of the signal digital frequencies is fmax, n0 is chosen as the following: With noise, the above estimation becomes only approxi- mate. (18) 1 (cid:20) (cid:20) fmax 2 We have the array manifold estimates from (10):
(cid:19)ek (cid:19)hk
(cid:19)ax k = (cid:19)qx ⎡
⎤
kk
(cid:19)fk =
⊗ (cid:19)gk = (cid:19)qx (cid:11) (cid:19)ek (cid:11) (cid:19)hk
(cid:19)qx 1 (cid:19)qx 1
=
Then we can get the unambiguous frequency estimates: (cid:21)(cid:13) arg , (19) Φt(cid:14) 2πn0 (23) .
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(cid:11) (cid:19)ek (cid:11) (cid:19)hk
(cid:10) θk, φk (cid:10) θk, φk (cid:10) θk, φk ... (cid:10) (cid:19)qx θk, φk M−1 (cid:10) (cid:19)qx θk, φk M−1
where arg{z} is principle argument of the complex number z between −π and π.
(2) Deriving the low-variance but ambiguous estimates of uk Define
(cid:10) θk, φk
(cid:11) (cid:19)ek,
(cid:10) θk, φk
(cid:11) (cid:19)hk,
(24) ci(k) = (cid:19)qx i bi(k) = (cid:19)qx i
Similarly, for the matrix pencil pair with spatial invari- ance along the x-axis, Ψx ’s eigenvalues equal {[Φx]kk = e j2π fkFSΔuk/c, k = 1, . . . , K }.
i = 1, 2, . . . , M − 1. Note that
(cid:9)
(cid:11)
(cid:9) × c∗
(cid:9) (cid:9) ×
Because Δ ≥ λk (k = 1, . . . , K) and −1 ≤ uk ≤1, there exists a set of cyclically related candidates for the estimation of uk [18]:
i (k) (cid:9) (cid:9) (cid:9) = (cid:9)ci(k)
(cid:11) (cid:19)ek (cid:11) (cid:19)ek
(cid:11) (cid:10) (cid:19)h∗ qx∗ θk, φk i k (cid:9) (cid:9) (cid:11) (cid:10) (cid:9) (cid:9)qx (cid:19)hk θk, φk i
(cid:10) nu
(cid:19)uk
= μk +
qx i (cid:9) (cid:9)qx i bi(k) (cid:9) (cid:9)bi(k) ,
=
(cid:10)
(cid:24) (cid:11)
(cid:26) (cid:11)
(cid:10)
(cid:9) (cid:9)
(cid:19)ek(cid:9) (cid:9)(cid:19)ek
(cid:10) θk, φk (cid:10) θk, φk (cid:19)h∗ k(cid:9) (cid:9) (cid:9) (cid:9)(cid:19)hk
− 1 − μk
− 1 − μk
(cid:23) (cid:19)fkFSΔ c
≤ nu ≤ (cid:21)(cid:13)
· c
M−1(cid:12)
(cid:9)
. , (20) (25) nuc (cid:19)fkFSΔ (cid:25) (cid:19)fkFSΔ c (cid:22) arg μk = , So we can get the estimate of Ponyting vector: Φx(cid:14) kk 2π (cid:19)fkFSΔ
k = 1 (cid:19)px
(cid:9) × c∗
i (k) (cid:9) (cid:9) (cid:9) . (cid:9)ci(k)
i=1
(26) M − 1 bi(k) (cid:9) (cid:9)bi(k) where (cid:7)x(cid:8) is the smallest integer not less than x; (cid:9)x(cid:10) is the largest integer not greater than x.
xk, (cid:19)px
yk, (cid:19)px
(3) Deriving the low-variance but ambiguous Unambiguous but high-variance estimates { (cid:19)px estimates of vk
zk} for {uk, vk, wk} have been achieved. This is the so-called vec- tor cross-product estimator who is pioneered by Nehorai and Paldi [1, 2] and firstly adapted to ESPRIT by Wong and Zoltowski [11, 24].
(cid:11)
(cid:10) nv
(cid:19)vk
= vk +
’s eigenvalues equal {[Φy Similarly, for the matrix pencil pair with spatial invari- ance along the y-axis, Ψy ]kk = e j2π fkFsΔvk/c, k = 1, . . . , K }. There exists a set of cyclically related candidates for the estimation of vk [18]: Similarly, for the matrix pencil with spatial invariance along the y-axis, we can get another set of unambiguous but y k for {uk, vk, wk}. For the matrix high-variance estimates (cid:19)p pencil with temporal invariance, we can get (cid:19)pt k.
(cid:24)
(cid:10)
(cid:11)
(cid:10)
− 1 − vk
(cid:26) (cid:11) ,
(cid:23) (cid:19)fkFSΔ c
(cid:22)
x j, (cid:19)px
· c
(5) Pairing the direction-cosine estimates and frequency estimates 1 − vk (21) nvc , (cid:19)fkFSΔ (cid:25) (cid:19)fkFSΔ c
yi, (cid:19)pt xk, (cid:19)p y
arg . vk =
≤ nv ≤ (cid:21)(cid:13) Φy(cid:14) kk 2π (cid:19)fkFSΔ
zi, i = 1, 2, . . . , K }, { (cid:19)px yk, (cid:19)p y xi, (cid:19)pt
y j, (cid:19)px z j, zk, k = 1, 2, . . . , K } are differ- zi} can be easily paired
yi, (cid:19)pt
The orderings of { (cid:19)pt xi, (cid:19)pt j = 1, 2, . . . , K } and { (cid:19)p y ent and need to be paired. { (cid:19)pt
100
x j, (cid:19)px
zk} as follows [18]:
yk, (cid:19)p y
) d a r (
y j, (cid:19)px (cid:28)
F. Ji and S. Kwong 5
(cid:14)
−
K
10−1
s e t a m
(cid:28)
(cid:14)
−
K
= arg min = arg min
i t s e
xk, (cid:19)p y z j } and { (cid:19)p y (cid:9) (cid:13) (cid:9) 1, . . . , (cid:19)pt (cid:19)pt (cid:9) (cid:13) (cid:9) 1, . . . , (cid:19)pt (cid:19)pt
with { (cid:19)px (cid:27)
(cid:13) (cid:19)px j1, . . . , (cid:19)px jK (cid:13) y y k1 , . . . , (cid:19)p (cid:19)p kK
e l g n a
10−2
(cid:14)(cid:9) (cid:9), (cid:14)(cid:9) (cid:9). (27) The above minimization is with respect to all possible per- mutations of {k1, . . . , kK } and { j1, . . . , jK }.
j0 1 , . . . , j0 K (cid:27) 1, . . . , k0 k0 K
k, (cid:19)px
k, (cid:19)p
10−3
⎡
y k we may form a (cid:19)pk: ⎤
y k
From (cid:19)pt
⎢ ⎣
⎥ ⎦ =
(cid:19)pk =
10−4
k + (cid:19)px (cid:19)pt k + (cid:19)p 3
(cid:19)pxk (cid:19)pyk (cid:19)pzk
f o n o i t a i v e d d r a d n a t s S M R
{ (cid:19)pt
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zi} are already paired with fi, and { (cid:19)px
yi, (cid:19)pt
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0
5
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xk, (cid:19)p y
SNR (dB)
xi, (cid:19)pt yk, (cid:19)p y z j } with μ j, { (cid:19)p y (cid:19)px (cid:19)fK } is to be paired with {(cid:19)μ j0
} and {(cid:19)vk0
K
K
x j, (cid:19)px y j, zk} with vk. It follows that { (cid:19)f1, . . . , 1 , . . . , (cid:19)μ j0 } [18].
1 , . . . , (cid:19)vk0
CRB From eigenvalue combined with eigenvectors From only eigenvectors
. (28)
(6) Disambiguation of the low-variance estimates of direction-cosine from ESPRIT’s eigenvalues [18]
(cid:10)
(cid:11)
(cid:11)
(cid:19)uk
The disambiguated estimates are
(cid:10) nv
(cid:19)vk
= μk +
= vk +
nu , , (29)
Figure 1: The RMS standard deviation of ( (cid:19)θk, (cid:19)φk, k = 1, 2) versus SNR: the two uncorrelated sources {θ1, θ2} = (30◦, 60◦), {φ1, φ2} = (40◦, −60◦), {γ1, γ2} = (0◦, 45◦), {η1, η2} = (0◦, 90◦), { f1, f2} = (0.3, 0.4) impinge upon an L-shaped vector sensor, 100 snapshots per experiment, 300 experiments per data point.
u and n◦
n◦ uc (cid:19)fkFSΔ n◦ vc (cid:19)fkFSΔ
10−1
s e t a m
where n◦
nu
i t s e
10−2
n◦ u = arg min
v may be separately estimated as (cid:20) (cid:20) (cid:20) (cid:20), (cid:20) (cid:20) (cid:20) (cid:20).
(cid:20) (cid:20) (cid:20) (cid:20) (cid:19)pxk − μk − (cid:20) (cid:20) (cid:20) (cid:20) (cid:19)pyk − vk −
nv
y c n e u q e r f
10−3
(30) n◦ v = arg min nuc (cid:19)fkFSΔ nvc (cid:19)fkFSΔ
10−4
(7) The 2D arrival angle estimation
10−5
(cid:29)(cid:30)
(cid:31)
(cid:19)θk = arcsin
We can calculate low-variance 2D arrival angle estimates from direction-cosine estimates out of ESPRIT’s eigenvalues
f o n o i t a i v e d d r a d n a t s S M R
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(cid:19)φk = arctan
k + (cid:19)v2 (cid:19)u2 k ! (cid:19)vk . (cid:19)uk
SNR (dB)
CRB Frequency estimates
, (31)
(cid:29)(cid:30)
(cid:19)θk = arcsin
(cid:31) ,
Similarly, we can calculate the high-variance 2D arrival angle estimates from direction-cosine estimates out of ES- PRIT’s eigenvectors
Figure 2: The RMS standard deviation of ( (cid:19)fk, k = 1, 2) versus SNR, same settings as an Figure 1.
(cid:19)φk = arctan
xk + (cid:19)p2 (cid:19)p2 yk ! (cid:19)pyk . (cid:19)pxk
(32)
Note that (cid:19)pzk may be applied to judge the quadrant of (cid:19)θk.
4. SIMULATIONS
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 spac- ing is chosen as Δ = 10 ∗ λmin/2 (λmin = c/( fmaxFs)) except for the example in Figures 4 and 5.
Several simulations are presented to verify the effectiveness of the proposed ESPRIT-based frequency and 2D angle estima- tion algorithm. In these simulations, the total-least-squares Figures 1 and 2 give the RMS standard deviations of ( (cid:19)θk, (cid:19)φk, k = 1, 2) and ( (cid:19)fk, k = 1, 2) versus SNR, respectively. The
100
10−1
) d a r (
10−1
s e t a m
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i t s e
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e l g n a
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10−3
s a i b S M R
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f o n o i t a i v e d d r a d n a t s S M R
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102
SNR (dB)
101 Intersensor space (λmin/2)
CRB From eigenvalue combined with eigenvectors From only eigenvectors
Angle estimates from only eigenvectors (rad) Frequency estimates Angle estimates from eigenvalue combined with eigenvectors (rad)
EURASIP Journal on Applied Signal Processing 6
Figure 3: The RMS bias of ( (cid:19)θk, (cid:19)φk, k = 1, 2) and ( (cid:19)fk, k = 1, 2) versus SNR, same settings as in Figure 1.
Figure 4: The RMS standard deviation of ( (cid:19)θk, (cid:19)φk, k = 1, 2) ver- sus intersensor spacing when SNR = 15 dB: the two uncorrelated sources {θ1, θ2} = (60◦, 30◦), {φ1, φ2} = (40◦, −60◦), {γ1, γ2} = (0◦, 45◦), {η1, η2} = (0◦, 90◦), { f1, f2} = (0.4, 0.5) impinge upon an L-shaped vector sensor, 100 snapshots per experiment, 300 experi- ments per data point.
10−2
) d a r (
10−3
s e t a m
i t s e
e l g n a
10−4
f o s a i b S M R
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100
102
101 Intersensor space (λmin/2)
From eigenvalue combined with eigenvectors From only eigenvectors
parameters of the two signals are {θ1, θ2} = (30◦, 60◦), {φ1, φ2} = (40◦, −60◦), {γ1, γ2} = (0◦, 45◦), {η1, η2} = (0◦, 90◦), { f1, f2} = (0.3, 0.4). Figure 3 gives the corresponding RMS bias versus SNR. The proposed algorithm successfully re- solves all the two electromagnetic source parameters includ- ing 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 5 dB 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.
Figure 5: The RMS bias of ( (cid:19)θk, (cid:19)φk, k = 1, 2) versus intersensor spac- ing when SNR = 15 dB, same settings as in Figure 4.
Figures 4 and 5, respectively, give the RMS standard de- viations and bias of ( (cid:19)θk, (cid:19)φk, k = 1, 2) versus intersensor spacing when SNR = 15. The parameters of the two sig- nals are {θ1, θ2} = (60◦, 30◦), {φ1, φ2} = (40◦, −60◦), {γ1, γ2} = (0◦, 45◦), {η1, η2} = (0◦, 90◦), { f1, f2} = (0.4, 0.5). Figure 3 shows that the standard deviations and bias of an- gle estimates from the eigenvalues combined with eigenvec- tors decrease as the intersensor spacing increases when Δ < 60λmin/2. But the performance of angle estimates obtained from only the eigenvectors remains relatively constant as the inter-sensor spacing increases. Note that when Δ ≥ 60λmin/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].
are the same as in Figure 1 except that { f1, f2} = (0.35, 0.4). One curve is calculated from the low-variance angle estima- tion algorithm when the signal frequencies are not known and estimated. Another curve is calculated by the low-var- iance angle estimation algorithm when the signal frequen- cies are known. It is shown that when signal frequencies are known, the RMS standard deviation of angle estimates is From (29), it can be seen that the performance of fre- quency estimation may affect the performance of low-var- iance angle estimation. Figure 6 gives the RMS standard devi- ation of ( (cid:19)θk, (cid:19)φk, k = 1, 2) versus SNR. The signal parameters
100
10−1
) d a r (
) d a r (
10−1
s e t a m
s e t a m
i t s e
i t s e
10−2
e l g n a
e l g n a
10−2
10−3
10−3
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f o n o i t a i v e d d r a d n a t s S M R
f o n o i t a i v e d d r a d n a t s S M R
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0
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20 Elevation angle of the first signal (deg)
SNR (dB)
From eigenvalue combined with eigenvectors From only eigenvectors
CRB Signal frequencies are known Signal frequencies are estimated
F. Ji and S. Kwong 7
Figure 6: The RMS standard deviations of ( (cid:19)θk, (cid:19)φk, k = 1, 2) ver- sus SNR from low-variance angle estimation, same settings as in Figure 1 except that { f1, f2} = (0.35, 0.4).
Figure 7: The RMS standard deviations of ( (cid:19)θk, (cid:19)φk, k = 1, 2) ver- sus elevation angle of the first signal when SNR = 15 dB. The pa- rameters of the two signals are θ2 = 45◦, {φ1, φ2} = (25◦, −30◦), {γ1, γ2} = (0◦, 45◦), {η1, η2} = (0◦, 90◦), { f1, f2} = (0.3, 0.4), 100 snapshots per experiment, 300 experiments per data point.
10−1
) d a r (
s e t a m
i t s e
10−2
e l g n a
10−3
just slightly lower than that when signal frequencies are esti- mated. Our simulations also show that RMS bias of the low- variance angle estimates when frequencies are known is al- most the same as that when frequencies are estimated.
f o n o i t a i v e d d r a d n a t s S M R
10−4
0
10
30
50
60
40
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90
20 70 Azimuth angle of the first signal (deg)
From eigenvalue combined with eigenvectors From only eigenvectors
Figure 7 gives the RMS standard deviation of ( (cid:19)θk, (cid:19)φk, k = 1, 2) versus elevation angle of the first signal when SNR = 15 dB. The parameters of the two signals are θ2 = 45◦, {φ1, φ2} = (25◦, −30◦), {γ1, γ2} = (0◦, 45◦), {η1, η2} = (0◦, 90◦), { f1, f2} = (0.3, 0.4). It is observed that the standard devia- tions 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 ( (cid:19)θk, (cid:19)φk, k = 1, 2) versus azimuth angle of the first signal when SNR = 15 dB. The signal parameters are the same as in Figure 7 ex- cept that {θ1, θ2} = (30◦, 45◦), φ2 = −30◦. It is shown that the RMS standard deviation of angle estimates from two esti- mation methods almost does not change as the azimuth an- gle of the first signal is changed.
Figure 8: The RMS standard deviation of ( (cid:19)θk, (cid:19)φk, k = 1, 2) versus azimuth angle of the first signal when SNR = 15 dB, same setting as in Figure 7 except that {θ1, θ2} = (30◦, 45◦) and φ2 = −30◦.
Figure 9 gives the RMS standard deviation of ( (cid:19)θk, (cid:19)φk, k = 1, 2) and ( (cid:19)fk, k = 1, 2) versus the number of snapshots when SNR = 15 dB. The parameters of the two signals are {θ1, θ2} = (60◦, 30◦), {φ1, φ2} = (40◦, −60◦), {γ1, γ2} = (0◦, 45◦), {η1, η2} = (0◦, 90◦), { f1, f2} = (0.3, 0.4). It is shown that the RMS standard deviation decreases slowly as the number of snapshots increases for the number of snap- shots exceeding 50.
It is observed that when Δ f 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. Figure 10 gives the RMS standard deviation and bias of ( (cid:19)fk, k = 1, 2) versus the difference Δ f of two signal fre- quencies when SNR = 15 dB. The signal parameters are the same as in Figure 9 except that { f1, f2} = (0.4 − Δ f , 0.4).
10−1
10−2
8 EURASIP Journal on Applied Signal Processing
10−3
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.
n o i t a i v e d d r a d n a t s S M R
10−4
ACKNOWLEDGMENTS
This work is supported by City University of Hong Kong Strategic Grant 7001697. This work is done when Fei Ji was visiting City University of Hong Kong.
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Number of snapshots
REFERENCES
[1] A. Nehorai and E. Paldi, “Vector sensor processing for electro- magnetic source localization,” in Proceedings of the 25th Asilo- mar Conference on Signals, Systems and Computers, vol. 1, pp. 566–572, Pacific Grove, Calif, USA, November 1991.
Angle estimates from only eigenvectors (rad) Frequency estimates Angle estimates from eigenvalue combined with eigenvectors (rad)
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Figure 9: The RMS standard deviation of ( (cid:19)θk, (cid:19)φk, k = 1, 2) and ( (cid:19)fk, k = 1, 2) versus the number of snapshots when SNR = 15 dB. The parameters of the two signals are {θ1, θ2} = (60◦, 30◦), {φ1, φ2} = (40◦, −60◦), {γ1, γ2} = (0◦, 45◦), {η1, η2} = (0◦, 90◦), { f1, f2} = (0.3, 0.4), 100 snapshots per experiment, 300 experi- ments per data point.
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Difference of two signal frequencies
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RMS standard deviation RMS bias
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Figure 10: The RMS standard deviations and bias of ( (cid:19)fk, k = 1, 2) versus the difference Δ f of two signal frequencies when SNR = 15 dB, same settings as in Figure 9 except that { f1, f2} = (0.4 − Δ f , 0.4), 50 snapshots per experiment, 300 experiments per data point.
5. CONCLUSION
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In this paper, we propose an ESPRIT-based algorithm that yields 2D angle and frequency estimates. This algorithm can
F. Ji and S. Kwong 9
