IEEE SENSORS JOURNAL, VOL. 12, NO. 6, JUNE 2012
1763
Various Compositions to Form a Triad of
Collocated Dipoles/Loops, for Direction
Finding and Polarization Estimation
Xin Yuan, Student Member, IEEE, Kainam Thomas Wong, Senior Member, IEEE, Zixin Xu, and Keshav Agrawal
Abstract— To form a collocated triad of orthogonally oriented
dipole(s) and/or loop(s), 20 different compositions are possible.
For each such composition: 1) closed-form formulas are produced
here to estimate the azimuth-elevation direction-of-arrival and
the polarization-parameters from an ambiguous steering vector
subject to an unknown complex-value multiplicative coefficient;
or 2) reasoning is given why such estimation is inviable.
Index Terms— Antenna arrays, aperture antennas, array signal
processing, direction of arrival estimation, diversity methods,
parameter estimation, polarization.
I. I NTRODUCTION
D
IVERSELY polarized antenna arrays have been much
investigated for direction finding, because polarization diversity provides an additional basis to resolve incident sources, besides on the basis of the sources’ different
directions-of-arrival and different time-frequency signal structures. For simultaneous estimation of the azimuth/elevation
arrival angles (φ, θ ) and polarization parameters (γ , η), a
minimum of three diversely polarized antennas are needed,
because at least three complex-value equations (based on three
antennas’ measurements) are needed to solved for the four
unknown real-value source-parameters (i.e. θ, φ, γ , η) and for
the one unknown complex-value multiplicative coefficient α,
which arises from the eigen-based estimation of the source’s
steering vector. Because at least three antennas are needed, this
paper will focus on diversely polarized antennas in a triad.
A collocated triad of diversely polarized antennas will be
investigated in this paper. For antennas spatially collocated in
a point-like geometry, no spatial phase-factor will exist among
them. These collocated antennas’ resulting array manifold will
thus enjoy these following advantages: (i) The array manifold
is independent from the spectrum of the incident signal. This
includes independence from the signal’s center-frequency and
Manuscript received August 27, 2011; revised October 6, 2011 and
December 2, 2011; accepted December 3, 2011. Date of publication
December 13, 2011; date of current version April 20, 2012. This work was
supported in part by the Internal Competitive Research under Grant G-U582,
awarded by the Hong Kong Polytechnic University. The Associate Editor
coordinating the review of this paper and approving it for publication was
Prof. Kiseon Kim.
X. Yuan and K. T. Wong are with the Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Kowloon 999077,
Hong Kong (e-mail: eiex.yuan@connect.polyu.hk; ktwong@ieee.org).
Z. Xu is with Ericsson Inc., Chengdu 100102, China.
K. Agrawal is with the Department of Electrical Engineering, University of
California, Los Angeles, CA 90024 USA.
Digital Object Identifier 10.1109/JSEN.2011.2179532
bandwidth.1 (ii) The array manifold is less sensitive to the
source’s distance from the antenna-array. (iii) In a K -source
scenario, the K azimuth-angle estimates, the K elevationangle estimates, and the K polarizations can all be intrinsically
associated to the proper sources, even without any further postestimation processing. [1] (iv) The antenna-triad is physically
more compact.
This work will investigate a collocated triad of dipole(s)
and/or loop(s). An electrically short dipole measures one
Cartesian component of the electric-field vector, along the
Cartesian axis on which the dipole is aligned. Similarly,
a magnetically small loop measures one Cartesian component of the magnetic-field vector, along which the loopaxis is aligned. Hence, with a collocated triad of three such
diversely polarized antennas, three components of the sixelement electromagnetic-field vector can be measured, at one
point in space.
To select three out of the six electromagnetic
components,
the number of different choices amount to 63 = 20. Among
these 20 possible configurations,
1) The dipole triad (a.k.a. a tripole) has been used for
direction-of-arrival estimation in [2]–[22], with closedform estimation formulas available in [6], [18], [19].
The dipole triad has also been used for polarization
estimation in [6], [18], with closed-form estimation
formulas available therein. For closed-form Cramér-Rao
bound expressions, please refer to [23]. For antennaelectromagnetic implementation of this tripole, please
see [13].
2) The loop triad has been used for direction-of-arrival
estimation in [6], [16], [18], [24], with closed-form
estimation formulas available in [6], [18]. The dipole
triad has also been used for polarization estimation in
[6], [18], with closed-form estimation formulas available
therein. For closed-form Cramér-Rao bound expressions,
please refer to [23]. For antenna-electromagnetic implementation of this loop triad, please see [24].
3) The {ex , e y , h z } triad has been investigated in [16];
however, no closed-form formula is available therein.
4) The {ez , h x , h y } triad has been investigated in
[25]–[29], with closed-form estimation formula available
for the azimuth arrival angle in [26], but no closedform estimation formula is available for the elevation
1 Real antenna patterns depend on frequency. The array manifold would not
depend on frequency if all antennas have the same frequency-dependence.
1530–437X/$26.00 © 2011 IEEE
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IEEE SENSORS JOURNAL, VOL. 12, NO. 6, JUNE 2012
arrival angle nor for the polarization parameters in any
of these references. For closed-form Cramér-Rao bound
expressions, please refer to [23].
5) The {ex , e y , h x } triad has been investigated in [16];
however, no closed-form formula is available therein.
In summary, only 2 out of 20 possible compositions have
closed-form formulas available in the open literature for the
estimation of the direction-of-arrival. Moreover, 15 of these
20 (i.e. other than 5 above) have been entirely overlooked
in the open literature. This paper aims to fill this literature
gap. Closed-form estimation-formulas for θ , φ, γ , and η (and
their associated validity regions) are herein advanced for the
first time in the open literature for 14 of the 18 overlooked
configurations. For the remaining 4 configurations, they will
be shown to be inadequate for such closed-form estimation.
II. T RIAD ’ S D IRECTIONAL AND P OLARIZATIONAL
R ESPONSES
A unit-power, completely polarized, transverse electromagnetic wave may be characterized by its six-component
electromagnetic-field vector, expressible in the Cartesian coordinates as [30]
⎡ ex ⎤
⎡ cos φ cos θ − sin φ ⎤
ey
cos θ
cos φ
e def ⎢ ez ⎥ def ⎢ sin−φsin
⎥ sin γ e j η
θ
0
(1)
h = ⎣ h x ⎦ = ⎣ − sin φ − cos φ cos θ ⎦
cos γ
hy
cos φ
− sin φ cos θ
hz
0
def
sin θ
=g
=
def
=
cos φ cos θ sin γ cos η−sin φ cos γ
sin γ cos η+cos φ cos γ
⎢ sin φ cos−θ sin
θ sin γ cos η
⎢
⎣ − sin φ sin γ cos η−cos φ cos θ cos γ
cos φ sin γ cos η−sin φ cos θ cos γ
sin θ cos γ
⎡
⎤
⎡ cos φ cos θ sin γ sin η ⎤
sin φ cos θ sin γ sin η
⎥
− sin θ sin γ sin η ⎥
⎥+ j⎢
⎣
⎦
− sin φ sin γ sin η ⎦
cos φ sin γ sin η
0
(2)
where θ ∈ [0, π] signifies the emitter’s elevation-angle
measured from the positive z-axis, φ ∈ [0, 2π) denotes
the corresponding azimuth-angle measured from the positive
x-axis, γ ∈ [0, π/2) refers to the auxiliary polarization angle,
and η ∈ [−π, π) symbolizes the polarization phase difference.
Note that depends on only the source’s direction-of-arrival,
whereas g depends on only the incident source’s polarization
state.
Any three of the six elements in (1) may be measured, at
a point in space, by a triad of collocated dipoles and/or loops
oriented along the corresponding Cartesian axes. Such a triad’s
3 × 1 array manifold would be
e
a=S
(3)
h
where S symbolizes a 3 × 6 selection-matrix of all zeroes,
except one entry of a “1” at a different position on each row.
For example, a dipole-triad has
⎡
⎤
100000
S = ⎣0 1 0 0 0 0⎦.
001000
III. C LOSED -F ORM F ORMULAS TO E STIMATE THE
A ZIMUTH -E LEVATION A RRIVAL A NGLES & THE
P OLARIZATION -PARAMETERS
In most eigen-based sensor-array parameter-estimation algorithms, an intermediate step estimates each incident source’s
steering vector, but correct to only within an unknown
complex-value scalar α. That is, available (in each algorithm
for each incident emitter)2 is the estimate â ≈ αa,3 from
which θ , φ, γ , and η are to be estimated. (This approximation becomes equality in noiseless or asymptotic cases.)
The question is whether â ≈ αa suffices to estimate θ ,
φ, γ , and η. The following five tables shows what, how,
and why.
Tables I to V list the closed-form estimation-formulas for
θ̂ , φ̂, γ̂ , η̂. These estimation-formulas are new to the open
literature, except as noted earlier in Section I, to the best
knowledge of the present authors. These formulas are valid,
except at a finite number of discrete values, which occur
with probability zero, anyway. Some estimation-formulas may
require some prior information as specified in the far-right
column in those tables. These formulas are obtained by algebraic and trigonometric manipulations of the three complexvalue equations (i.e. six real-value equations) from the three
component-antenna consisting of the triad, to solve for the six
real-value unknown scalars of θ , φ, γ , η, Re{α}, and Im{α}.
The Appendix shows such detailed algebraic and trigonometric
manipulations for the composition of {ex , e y , h z }. The derivation would be similar for the other compositions. In these
estimation-formulas, Re{·} refers to the real-value part of the
entity inside the curly brackets, Im{·} denotes to the imaginaryvalue part of the entity inside the curly brackets, {·} signifies
the complex-phase angle of the entity inside the curly brackets,
[·] represents the th element of the vector inside the square
brackets, and · symbolizes the Frobenius norm of the vector
inside.
Below are some qualitative observations on the above
estimation-formulas:
2 This does NOT presume only a single incident source. Multiple, possibly
cross-correlated/broadband/time-varying sources can be handled. For details,
please refer to those references directly.
3 The following simple signal-and-noise model demonstrates how eigenbased parameter-estimation algorithms would lead to â ≈ αa.
Suppose the emitted signal arrives at the triad as s(t), but corrupted
additively by the triad’s thermal noise-vector n(t). The triad’s measured data
thus equals a 3 × 1 vector z(t) = s(t)a + n(t), at each t = tm . From M such
time-samples, an eigen-based parameter-estimation
algorithm can form a 3×3
1 M z(t ) [z(t )] H , where the superscript
data covariance matrix Ĉ = M
m
m
m=1
H symbolizes the hermitian operator. Suppose further that {s(t), ∀t} and
{n(t), ∀t} are each temporally white, each temporally stationary, and not crosscorrelated. Then, Ĉ ≈ C = Ps aa H + Pn I, where Ps denotes the power of
the incident signal, Pn refers to the thermal noise power at each antenna,
and I signifies a 3 × 3 identity matrix. This 3 × 3 matrix Ĉ is Hermitian,
and asymptotically approaches C, as M → ∞. The asymptotic C has a
principal eigenvector equal to αa, where α can be any complex-value number
that has a magnitude of 1/a and that is algebraically independent of a.
Hence, the principal eigenvector of the sampled data-covariance matrix Ĉ is
approximately αa.
This â ≈ αa could arise also for more complex data-models involving
multiple incident sources, more complicated channels, and more complicated
noises. However, this â ≈ αa may be unobtainable for some signals/noise
models, especially where two or more incident sources have closely aligned
steering vectors and thus not individually resolvable.
YUAN et al.: VARIOUS COMPOSITIONS TO FORM A TRIAD OF COLLOCATED DIPOLES/LOOPS
1765
TABLE I
F ORMULAS FOR D IRECTION F INDING & P OLARIZATION E STIMATION , A LREADY AVAILABLE IN THE L ITERATUREa
Composition
1.1
Antennas
{ex , e y , ez }
[6], [18]
Estimation formulas
⎧
⎪
−[d]3
⎪
⎪
, if (Re{[d]1 } cos φ̂ + Re{[d]2 } sin φ̂) ≤ 0
⎨ tan−1
Re{[d]1 } cos φ̂+Re{[d]2 } sin φ̂
θ̂ =
⎪
−[d]3
⎪
⎪
+ π, if (Re{[d]1 } cos φ̂ + Re{[d]2 } sin φ̂) > 0
⎩ tan−1
Re{[d]1 } cos φ̂+Re{[d]2 } sin φ̂
⎧
⎪
−
Im
{[d]
}
⎪
⎪
⎨ tan−1 Im{[d] 1} , if (sin η · Im{[d]2 }) ≥ 0
2
φ̂ =
⎪
−Im{[d] }
⎪
⎪
⎩ tan−1 Im{[d] 1} + π, if (sin η · Im{[d]2 }) < 0
Intermediate variables
− j [â]
def
d = âe â 3
Prior info. required
η ∈ [0, π )
or η ∈ [−π, 0)
2
1.2
{h x , h y , h z }
[6], [18]
|[d]3 |
γ̂ = sin−1
sin θ̂
η̂ = − [d]1 sin φ̂ − [d]2 cos φ̂
⎧
⎪
−[d]3
⎪
⎪
, if (Re{[d]1 } cos φ̂ + Re{[d]2 } sin φ̂) ≤ 0
⎨ tan−1
Re{[d]1 } cos φ̂+Re{[d]2 } sin φ̂
θ̂ =
⎪
−[d]
⎪
3
⎪
+ π, if (Re{[d]1 } cos φ̂ + Re{[d]2 } sin φ̂) > 0
⎩ tan−1
Re{[d]1 } cos φ̂+Re{[d]2 } sin φ̂
⎧
⎪
−Im{[d] }
⎪
⎪
⎨ tan−1 Im{[d] 1} , if (sin η · Im{[d]2 }) ≥ 0
2
φ̂ =
⎪
−Im{[d]1 }
⎪
−1
⎪
⎩ tan
Im{[d] } + π, if (sin η · Im{[d]2 }) < 0
1.3
1.4
[26], [28]
{ex , e y , h z }
{ez , h x , h y }
|[d] |
− j [â]
def
d = âe â 3
η ∈ [0, π )
or η ∈ [−π, 0)
2
3
γ̂ = cos−1
sin θ̂
η̂ = [d]2 cos φ̂ − [d]1 sin φ̂
⎧
⎪
Re [b]3
⎪ −1 ⎪
⎨ sin
Re [b] cos φ̂−Re[b] sin φ̂ , if θ ∈ [0, π/2]
2
1
θ̂ =
⎪
Re [b]3
⎪
⎪ π − sin−1 ⎩
Re [b] cos φ̂−Re[b] sin φ̂ , if θ ∈ (π/2, π ]
2
1
⎧
⎪
⎪
⎨ tan−1 Im{[b]2 } , if (cos θ sin η · Im{[b]1 }) ≥ 0
Im{[b]1 }
φ̂ =
⎪
⎪
⎩ tan−1 Im{[b]2 } + π, if (cos θ sin η · Im{[b]1 }) < 0
Im{[b]1 }
Im{[b] } tan θ̂ 1
γ̂ = tan−1 Re{[b]3 } cos φ̂ sin η̂ [b]1 cos φ̂+[b]2 sin φ̂
η̂ = cos θ̂
⎧
⎪
Re [e]1
⎪
, if θ ∈ [0, π/2].
⎪
⎨ sin−1 Re [e]2 sin φ̂−Re [e]3 cos φ̂ θ̂ =
⎪
Re [e]1
⎪
, if θ ∈ (π/2, π ].
⎪
⎩ π − sin−1 Re [e]2 sin φ̂−Re [e]3 cos φ̂ ⎧
⎪
Im{[e] }
⎪
⎪
⎨ tan−1 Im{[e]3 } , if (cos θ sin η · Im{[e]2 }) ≤ 0
2
φ̂ =
⎪
Im{[e] }
⎪
⎪
⎩ tan−1 Im{[e]3 } + π, if (cos θ sin η · Im{[e]2 }) > 0
2
Re{[e] } sin φ̂ sin η̂ 1
γ̂ = tan−1 Im{[e]3 } tan θ̂ [e]2 cos φ̂+[e]3 sin φ̂
η̂ = −
θ ∈ [0, π2 ]
or θ ∈ ( π2 , π ]
b = âe− j [â]3
def
η ∈ [0, π )
or η ∈ [−π, 0)
θ ∈ [0, π2 ]
or θ ∈ ( π2 , π ]
def
e = âe− j [â]1
η ∈ [0, π )
or η ∈ [−π, 0)
cos θ̂
a For compositions 1.1 and 1.2, the φ̂ formula in [6] is erroneous.
1) The various compositions are organized into the five
tables, largely according to the prior knowledge needed
in these closed-form estimation-formulas: Based on the
prior requirement knowledge, the 16 configurations can
be classified into 4 different groups:
a) The all-dipole or all-loop triads (i.e. compositions 1.1 and 1.2): Some prior knowledge of only
η is required, to resolve a π-ambiguity in φ̂ as
shown in Table I.
b) Triads with both the z-oriented dipole and the
z-oriented loop (i.e. compositions 2.1 to 2.4):
Some prior knowledge of only θ is required, to
resolve a π-ambiguity in φ̂ as shown in Table II.
c) Other triads that has exactly one component-
antenna along each of the three Cartesian axes
(i.e. compositions 3.1 to 3.4): Some prior
knowledge of only φ is required, to resolve a
π-ambiguity in φ̂ as shown in Table III.
d) The remaining 6 compositions (i.e. compositions
1.3-1.4 in Table I, and compositions 4.1-4.4 in
Table IV and V): Some prior knowledge of both
θ and η is required, to resolve π-ambiguities in
φ̂, θ̂ , and η̂.
2)
a) The dipole-triad {ex , e y , ez } and the loop-triad
{h x , h y , h z } each has a validity region for (θ, φ)
covering the entire sphere, if the sign of sin η
is known a priori. This is because the dipoletriad {ex , e y , ez } and the loop-triad {h x , h y , h z },
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IEEE SENSORS JOURNAL, VOL. 12, NO. 6, JUNE 2012
TABLE II
F ORMULAS FOR D IRECTION F INDING & P OLARIZATION E STIMATION , FOR T RIADS WITH B OTH THE z-D IPOLE AND THE z-L OOP
Composition
Antennas
Estimation formulas
⎧
⎪
⎪
⎨
θ̂ =
2.1
2.2
2.3
2.4
{ex , ez , h z }
{e y , ez , h z }
{ez , h x , h z }
{ez , h y , h z }
⎧
⎪
⎪
⎪
⎪
−1
⎪
⎪
⎪ tan
⎪
⎪
⎪
⎪
⎩
⎪
⎨
Intermediate variables
⎫
!
⎪
⎬
1−D12 −D22 + D14 +(D22 −1)2 +2D12 (1+D22 ) ⎪
√
,
⎪
2|D1 |
⎪
⎭
if
θ ∈ [0, π/2].
⎧
⎫
!
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 1−D 2 −D 2 + D 4 +(D 2 −1)2 +2D 2 (1+D 2 ) ⎪
⎬
⎪
⎪
1
1
2
2
1
2
−1
⎪
√
⎪
, if θ
π − tan
⎪
⎪
⎪
⎪
2|D1 |
⎩
⎪
⎪
⎩
⎭
⎧
⎪
⎪ tan−1 D2 cos θ̂ ,
if D1 tan θ̂ < 0,
⎨
−D1
φ̂ =
⎪
D2 cos θ̂
⎪
−1
⎩ tan
+ π, if D1 tan θ̂ ≥ 0.
−D1
[â] 2
γ̂ = arctan [â]
3
[â]2
η̂ = −π
[â]3
⎧
⎫
⎧
!
⎪
⎪ 1−D 2 −D 2 + D 4 +(D 2 −1)2 +2D 2 (1+D 2 ) ⎪
⎪
⎪
⎨
⎬
⎪
⎪
2
1
2
2
1
1
⎪
−1
⎪
√
tan
,
if θ
⎪
⎪
⎪
⎪
⎪
2|D1 |
⎪
⎪
⎪
⎩
⎭
⎪
⎨
θ̂ =
⎫
⎧
!
⎪
⎪
⎪
⎪
⎪
⎪ 1−D 2 −D 2 + D 4 +(D 2 −1)2 +2D 2 (1+D 2 ) ⎪
⎪
⎬
⎨
⎪
⎪
1
2
1
2
1
2
⎪
−1
√
⎪ π − tan
, if θ
⎪
⎪
⎪
⎪
2|D1 |
⎩
⎪
⎪
⎭
⎩
⎧
D1
⎪
−1
⎪
if D2 < 0,
,
⎨ tan
D2 cos θ̂ φ̂ =
D
⎪
1
⎪
+ π, if D2 ≥ 0.
⎩ tan−1
D2 cos θ̂
[â] 2
γ̂ = arctan [â]
3
[â]2
η̂ = −π
[â]3
⎧
⎫
⎧
!
⎪
⎪
⎪ 1−D 2 −D 2 + D 4 +(D 2 +1)2 +2D 2 (−1+D 2 ) ⎬
⎪
⎪
⎨
⎪
⎪
1
2
1
2
1
2
⎪
⎪ tan−1
√
,
if
⎪
⎪
⎪
⎪
⎪
2|D
|
⎪
⎪
⎪
2
⎩
⎪
⎭
⎨
θ̂ =
⎧
⎫
!
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 1−D 2 −D 2 + D 4 +(D 2 +1)2 +2D 2 (−1+D 2 ) ⎪
⎬
⎪
⎪
2
1
2
2
1
1
⎪
√
⎪
, if
π − tan−1
⎪
⎪
⎪
⎪
2|D
|
⎩
⎪
⎪
2
⎩
⎭
⎧
D1 cos θ̂
⎪
⎪
,
if D2 tan θ̂ ≥ 0,
⎨ tan−1
D2
φ̂ =
⎪
D1 cos θ̂
−1
⎪
⎩ tan
+ π, if D2 tan θ̂ < 0.
D2
[â] γ̂ = arctan 1 [â]
3
[â]1
η̂ = −π
[â]3
⎧
⎫
⎧
!
⎪
⎪
⎪
⎪
⎨ 1−D 2 −D 2 + D 4 +(D 2 +1)2 +2D 2 (−1+D 2 ) ⎪
⎬
⎪
⎪
1
2
1
2
1
2
⎪
−1
⎪
√
,
if
⎪ tan
⎪
⎪
⎪
⎪
2|D2 |
⎪
⎪
⎪
⎩
⎪
⎭
⎨
θ̂ =
⎫
⎧
!
⎪
⎪
⎪
⎪
⎪
⎪ 1−D 2 −D 2 + D 4 +(D 2 +1)2 +2D 2 (−1+D 2 ) ⎪
⎪
⎬
⎨
⎪
⎪
1
1
2
2
1
2
−1
⎪
√
⎪ π − tan
, if
⎪
⎪
⎪
⎪
2|D2 |
⎩
⎪
⎪
⎭
⎩
⎧
−D2
⎪
⎪ tan−1
if D1 < 0,
,
⎨
D1 cos θ̂ φ̂ =
−D
⎪
−1
2
⎪
+ π, if D1 ≥ 0.
⎩ tan
D 1 cos θ̂
[â] 1
γ̂ = arctan [â]
3
[â]1
−π
η̂ = [â]3
but no other composition, enjoys one additional
constraint from the normalization e = 1 or
h = 1.
b) For the four compositions of {ex , ez , h y },
{ex , h y , h z }, {e y , ez , h x }, {e y , h x , h z }, the φ̂ suffers ambiguity due to the presence of the cos−1
or sin −1 functions.
c) For the four compositions with the z-oriented
dipole and the z-oriented loop, the azimuthelevation direction-of-arrival estimate has (θ, φ)
validity region over only an hemisphere. This
3)
Prior info.
required
∈ (π/2, π ].
def
b = âe− j [â]3
Im{[b]1 }
D1 =
&
%Im{[b]2 }
Re{[b]1 }
− D1 tan γ̂ cos η̂
D2 =
Re{[b]2 }
or
θ ∈ [0, π2 ]
θ ∈ ( π2 , π ]
∈ [0, π/2].
∈ (π/2, π ].
def
b = âe− j [â]3
Im{[b]1 }
D1 = Im{[b] }
2
&
%
Re{[b]1 }
− D1 tan γ̂ cos η̂
D2 =
Re{[b]2 }
θ ∈ [0, π2 ]
or θ ∈ ( π , π ]
2
θ ∈ [0, π/2].
θ ∈ (π/2, π ].
def
b = âe− j [â]3
Im{[b]2 }
D1 =
%Im{[b]1 }
&
Re{[b] }
D2 = Re{[b]2 } − D1 tan γ̂ cos η̂
1
or
θ ∈ [0, π2 ]
θ ∈ ( π2 , π ]
θ ∈ [0, π/2].
θ ∈ (π/2, π ].
def
b = âe− j [â]3
Im{[b]2 }
D1 =
&
%Im{[b]1 }
Re{[b]2 }
− D1 tan γ̂ cos η̂
D2 =
Re{[b]1 }
or
θ ∈ [0, π2 ]
θ ∈ ( π2 , π ]
is because θ̂ can be estimated only to tan2 θ̂ ,
hence there is ambiguity between θ ∈ [0, π2 ] or
θ ∈ ( π2 , π].
d) Similarly for the six remaining compositions, φ̂
can be estimated only to within an hemisphere,
and requires sin η. The parameter η refers to
the phase by which the electric field’s y-axis
component leads the x-axis component.
a) For those compositions with exactly one
component-antenna along each of x-, y-,
and z-axis, the azimuth-elevation direction-
YUAN et al.: VARIOUS COMPOSITIONS TO FORM A TRIAD OF COLLOCATED DIPOLES/LOOPS
1767
TABLE III
F ORMULAS FOR D IRECTION F INDING & P OLARIZATION E STIMATION , FOR O THER T RIADS WITH A
C OMPONENT-A NTENNA A LONG E ACH C ARTESIAN A XIS
Composition
Antennas
3.1
{ex , ez , h y }
Estimation formulas
Im{[c]3 }
θ̂ = cos−1 Im
{[c]1 }
⎧
⎪
Re{[c]1 } cos θ̂ −Re{[c]3 } , if sin φ ≥ 0,
⎪
⎨ cos−1
Re{[c]2 } sin θ̂
φ̂ =
⎪
{[c]1 } cos θ̂ −Re{[c]3 }
Re
⎪
−1
⎩ − cos
, if sin φ < 0.
Re{[c]2 } sin θ̂
[c]2 sin φ̂
γ̂ = tan−1 [c]1 sin θ̂+[c]2 cos φ̂ cos θ̂ [c]2 sin φ̂
η̂ = Intermediate variables
def
c = âe− j [â]2
Prior info. required
φ ∈ [0, π )
or φ ∈ [π, 2π )
[c]1 sin θ̂ +[c]2 cos φ̂ cos θ̂
Im{[b]1 }
Im
{[b]2 }
⎧
⎪
Re{[b]2 } cos θ̂−Re{[b]1 } , if cos φ ≥ 0,
⎪
⎨ sin−1
Re{[b]3 } sin θ̂
φ̂ =
⎪
⎪
⎩ π − sin−1 Re{[b]2 } cos θ̂−Re{[b]1 } , if cos φ < 0.
Re{[b]3 } sin θ̂
[b] sin θ̂ +[b] sin φ̂ cos θ̂ −1
2
3
γ̂ = tan
[b]3 cos φ̂
[b]
sin
θ̂+[b]
sin
φ̂
cos
θ̂
2
3
η̂ = θ̂ = cos−1
3.2
{ex , h y , h z }
3.3
{e y , ez , h x }
def
b = âe− j [â]3
φ ∈ [− π2 , π2 )
or φ ∈ [ π2 , 3π
2 )
[b]3 cos φ̂
θ̂ = cos−1
⎧
⎪
Re{[c]3 }+Re{[c]1 } cos θ̂ , if cos φ ≥ 0,
⎪
⎨ sin−1
Re{[c]2 } sin θ̂
φ̂ =
⎪
{[c]3 }+Re{[c]1 } cos θ̂
Re
⎪
⎩ π − sin−1
, if cos φ < 0.
Re{[c]2 } sin θ̂
−[c]2 cos φ̂
γ̂ = tan−1 [c]1 sin θ̂+[c]2 sin φ̂ cos θ̂ −[c]2 cos φ̂
η̂ = −Im{[c]3 }
Im{[c]1 }
def
c = âe− j [â]2
φ ∈ [− π2 , π2 )
or φ ∈ [ π2 , 3π
2 )
[c]1 sin θ̂ +[c]2 sin φ̂ cos θ̂
θ̂ = cos−1
3.4
{e y , h x , h z }
−Im{[b]1 }
Im{[b]2 }
⎧
⎪
Re{[b]1 }+Re{[b]2 } cos θ̂ , if sin φ ≥ 0,
⎪
⎨ cos−1
Re{[b]3 } sin θ̂
φ̂ =
⎪
{[b]1 }+Re{[b]2 } cos θ̂
Re
⎪
−1
⎩ − cos
, if sin φ < 0.
Re{[b]3 } sin θ̂
γ̂ = tan−1 [b]2 sin θ̂ +[b]3 cos φ̂ cos θ̂ −[b]3 sin φ̂
[b]
sin
θ̂+[b]
2
3 cos φ̂ cos θ̂
η̂ = def
b = âe− j [â]3
φ ∈ [0, π )
or φ ∈ [π, 2π )
−[b]3 sin φ̂
of-arrival needs be estimated before the
polarization parameters. There, the resulting
array manifold’s x-component and y-component
would have imaginary parts interrelated by either
tan φ or cos θ . For example, the composition
{ex , e y , ez } has an x-component with an
imaginary part of Im{− cos(θ ) cos(φ) sin(γ ) +
cos(γ ) sin(φ) cos(η) − j cos(γ ) sin(φ) sin(η)},
and a y-component with an imaginary
part
of
Im{− cos(θ ) sin(φ) sin(γ )
−
cos(γ ) cos(φ) cos(η) + j cos(γ ) cos(φ) sin(η)}.
They are interrelated by − tan φ.
b) For all other compositions, the polarization parameters need to be estimated before the azimuthelevation direction-of-arrival. For example, for
compositions with both the z-axis dipole and the
z-axis loop, their array manifolds have entries
e j η sin(θ ) sin(γ ) and sin(θ ) cos(γ ), interrelated
by a ratio of −e j η tan(γ ), which depends on
only the polarization parameters.
IV. W HY C LOSED -F ORM E STIMATION -F ORMULA
Un AVAILABLE FOR THE OTHER F OUR C OMPOSITIONS
Closed-form estimation-formulas are given in Section III
for 16 compositions (out of 20 possible compositions) of a
triad of orthogonally oriented dipole(s) and/or loop(s). The
commonality among those 16 compositions are their inclusion
of a z-axis dipole and/or a z-axis loop. Without at least one
of these two, no closed-form estimation-formula is possible,
due to the under-determined condition explained below.
Recall that each triad provides six real-value equations, from which the six real-value scalar unknowns of
1768
IEEE SENSORS JOURNAL, VOL. 12, NO. 6, JUNE 2012
TABLE IV
F ORMULAS FOR D IRECTION F INDING AND P OLARIZATION E STIMATION , FOR O THER T RIADS WITH A z-A XIS D IPOLE OR L OOP
Composition Antennas
Estimation formulas
θ̂ =
φ̂ =
4.1
{ex , ez , h x }
γ̂ =
η̂ =
θ̂ =
φ̂ =
4.2
{ex , h x , h z }
γ̂ =
η̂ =
Intermediate variables
⎧
⎧
⎫
!
⎪
⎪
⎪
⎨ 1−D32 −D42 + D34 +(D42 +1)2 +2D32 (−1+D42 ) ⎪
⎬
⎪
⎪
⎪
√
tan−1
, if θ ∈ [0, π/2].
⎪
⎪
2|D4 |
⎪
⎪
⎪
⎨
⎩
⎭
⎧
⎫
!
⎪
⎪
⎪
⎨ 1−D32 −D42 + D34 +(D42 +1)2 +2D32 (−1+D42 ) ⎪
⎬
⎪
⎪
−1
⎪
√
⎪
, if θ ∈ (π/2, π ].
⎪ π − tan ⎪
⎪
2|D
|
⎪
4
⎩
⎩
⎭
⎧
⎪
⎪
D3 cos θ̂
⎪
, if D4 tan θ̂ ≥ 0.
⎨ tan−1
D4
def
c = âe− j [â]2
⎪
⎪
D3 cos θ̂
−1
⎪
⎩ tan
+ π, if D4 tan θ̂ < 0.
D4
−Im{[c]1 } sin γ̂
D3 = Re{[c] } cos
γ̂ sin η̂
2
{[c]
}
Im
{[c]
}−
Re
{[c]
}
Im
{[c]
}
|
|Re
1
3
3
1
tan−1 '
−
Im
{[c]
}
sin
γ̂
2
2
2
2
3
}+Re{[c]3 }Im{[c]3 })
D4 = Re{[c] } cos
⎧ ⎧ (Im{[c]3 } +Im{[c]1 } ) +(Re{[c]1 }Im{[c]1⎫
γ̂ sin η̂
2
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎨ −(Im{[c]3 }Re{[c]3 } + Im{[c]1 }Re{[c]1 }) ⎪
⎪
⎪
⎪
⎪
, if sin η ≥ 0,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⎩ + j (Im{[c] }2 + Im{[c] }2 )
⎨ ⎪
3
1
⎧
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
−(Im{[c]
}Re{[c]
}
+
Im{[c]
}Re{[c]
})
3
3
1
1 ⎬
⎪
⎪
⎪
+ π, if sin η < 0.
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩ + j (Im{[c] }2 + Im{[c] }2 )
⎭
⎩ ⎪
3
1
⎧
⎫
⎧
!
⎪
2
2
4
⎪
⎨ 1−D5 −D6 + D5 +(D62 +1)2 +2D52 (−1+D62 ) ⎪
⎬
⎪
⎪ −1
⎪
√
⎪
tan
, if θ ∈ [0, π/2].
⎪
⎪
2|D
|
⎪
⎪
6
⎪
⎩
⎭
⎨
⎧
⎫
!
⎪
⎪
⎪
⎨ 1−D52 −D62 + D54 +(D62 +1)2 +2D52 (−1+D62 ) ⎪
⎬
⎪
⎪
⎪
⎪
√
, if θ ∈ (π/2, π ].
π − tan−1
⎪
⎪
2|D
|
⎪
⎪
⎩
6
⎩
⎭
⎧
⎪
⎪
D5 cos θ̂
⎪
,
if D6 tan θ̂ ≥ 0.
⎨ tan−1
D6
def
b = âe− j [â]3
⎪
⎪
D5 cos θ̂
⎪
⎩ tan−1
+ π, if D6 tan θ̂ < 0.
−Im{[b]2 } cos γ̂
D6
D5 = Re
{[b]3 } sin γ̂ sin η̂
'
2
2
2
2
(Im{[b]1 } +Im{[b]2 } ) +(Re{[b]1 }Im{[b]1 }+Re{[b]2 }Im{[b]2 })
tan−1
{[b] } cos γ̂
Im
|Re{[b]1 }Im{[b]2 }−Re{[b]2 }Im{[b]1 }|
D6 = Re{[b] } 1sin γ̂ sin η̂
⎧ ⎧
⎫
3
⎪
⎪
⎪
⎪
⎪
⎪ ⎪
⎨
⎬
(Re{[b]
}Im{[b]
}
+
Re{[b]
}Im{[b]
})
⎪
1
1
2
2
⎪
⎪
⎪
, if sin η ≥ 0,
⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎩ + j (Im{[b]1 }2 + Im{[b]2 }2 )
⎭
⎨ ⎪
⎧
⎫
⎪
⎪
⎪ (Re{[b] }Im{[b] } + Re{[b] }Im{[b] }) ⎪
⎪
⎪ ⎪
⎨
⎪
1
1
2
2 ⎬
⎪
⎪
⎪
+ π, if sin η < 0.
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩ + j (Im{[b] }2 + Im{[b] }2 )
⎭
⎩ ⎪
1
θ ∈ [0, π2 ]
or θ ∈ ( π2 , π ]
η ∈ [0, π )
or η ∈ [−π, 0)
θ ∈ [0, π2 ]
or θ ∈ ( π2 , π ]
η ∈ [0, π )
or η ∈ [−π, 0)
2
θ, φ, γ , η, Re{α}, Im{α} are to be determined. However, these
six equations would be linearly dependent for any of the
four compositions without a z-axis dipole and without
z-axis loop. Hence, the six unknowns would be underdetermined.
For example, the triad {ex , e y , h x } has these six linearly
dependent equations:
Re{α}(cos φ cos θ sin γ cos η − sin φ cos γ )
−Im{α}(cos φ cos θ sin γ sin η) = Re{[â]1 }
(4)
Re{α}(sin φ cos θ sin γ cos η + cos φ cos γ )
−Im{α}(sin φ cos θ sin γ sin η) = Re{[â]2 }
(5)
Re{α}(− sin φ sin γ cos η − cos φ cos θ cos γ )
−Im{α}(− sin φ sin γ sin η) = Re{[â]3 }
Prior info.
required
Re{α}(cos φ cos θ sin γ sin η) + Im{α}
(cos φ cos θ sin γ cos η − sin φ cos γ ) = Im{[â]1 }
(7)
Re{α}(sin φ cos θ sin γ sin η) + Im{α}
(sin φ cos θ sin γ cos η + cos φ cos γ ) = Im{[â]2 }
Re{α}(− sin φ sin γ sin η) + Im{α}
(8)
(− sin φ sin γ cos η − cos φ cos θ cos γ ) = Im{[â]3 }. (9)
These six equations are linearly dependent, as
(Im{α}) × (5) = (Re{α}) × (8) − (Re{α})(tan φ) × (7)
(6)
+(Im{α})(tan φ) × (4)
(Im{α}) × (5) = (Re{α}) × (8) + (Re{α})(cos θ ) × (9)
−(Im{α})(cos θ ) × (6)
YUAN et al.: VARIOUS COMPOSITIONS TO FORM A TRIAD OF COLLOCATED DIPOLES/LOOPS
1769
TABLE V
F ORMULAS FOR D IRECTION F INDING & P OLARIZATION E STIMATION , FOR O THER T RIADS WITH A z-A XIS D IPOLE OR L OOP
Composition
Antennas
Estimation Formulas
θ̂ =
4.3
4.4
{e y , ez , h y }
{e y , h y , h z }
⎧
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
−1
⎪
tan
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎨
Intermediate Variables
⎫
!
⎪
⎬
1−D72 −D82 + D74 +(D82 +1)2 +2D72 (−1+D82 ) ⎪
√
,
⎪
2|D8 |
⎪
⎭
if
θ ∈ [0, π/2].
⎧
⎫
!
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 1−D 2 −D 2 + D 4 +(D 2 +1)2 +2D 2 (−1+D 2 ) ⎪
⎬
⎪
⎪
7
8
7
8
7
8
⎪
−1
√
⎪
, if θ ∈ (π/2, π ].
⎪ π − tan
⎪
⎪
⎪
2|D
|
⎩
⎪
⎪
8
⎩
⎭
⎧
D8
⎪
⎪ tan−1
,
if D7 ≥ 0.
⎨
D
cos
θ̂
7
φ̂ =
⎪
D8
−1
⎪
⎩ tan
+ π, if D7 < 0.
⎫
⎧ D7 cos θ̂
⎬
⎨
|Re{[c]1 }Im{[c]3 }−Re{[c]3 }Im{[c]1 }|
γ̂ = tan−1 !
⎩ (Im{[c] }2 +Im{[c] }2 )2 +(Re{[c] }Im{[c] }+Re{[c] }Im{[c] })2 ⎭
3
1
1
1
3
3
⎫
⎧ ⎧
⎪
⎬
⎨ −(Im{[c] }Re{[c] } + Im{[c] }Re{[c] }) ⎪
⎪ ⎪
⎪
3
3
1
1
⎪
⎪
,
if sin η ≥ 0,
⎪
⎪
⎪
⎪
⎭
⎩ + j (Im{[c] }2 + Im{[c] }2 )
⎨ ⎪
3
1
⎧
⎫
η̂ =
⎪ ⎨
⎪
⎪
⎪
⎪
−(Im{[c]3 }Re{[c]3 } + Im{[c]1 }Re{[c]1 }) ⎬
⎪
⎪
⎪
+ π, if sin η < 0.
⎪
⎪
⎪
⎩ ⎪
⎩ + j (Im{[c] }2 + Im{[c] }2 )
⎭
3
1
⎧
⎫
⎧
!
⎪
⎪
⎪
⎪
⎨ 1−D 2 −D 2 + D 4 +(D 2 +1)2 +2D 2 (−1+D 2 ) ⎪
⎬
⎪
⎪
9
9
10
10
9
10
⎪
−1
⎪
√
,
if θ ∈ [0, π/2].
⎪
⎪ tan
⎪
⎪
⎪
2|D
|
⎪
⎪
⎪
10
⎩
⎪
⎭
⎨
θ̂ =
⎫
⎧
!
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎨ 1−D 2 −D 2 + D 4 +(D 2 +1)2 +2D 2 (−1+D 2 ) ⎪
⎪
⎪
9
10
9
10
9
10
⎪
√
⎪
, if θ ∈ (π/2, π ].
π − tan−1
⎪
⎪
⎪
⎪
2|D10 |
⎩
⎪
⎪
⎭
⎩
⎧
D10
⎪
⎪ tan−1
if D9 ≥ 0.
,
⎨
D
cos
θ̂
9
φ̂ =
⎪
D10
−1
⎪
⎩ tan
+ π, if D9 < 0.
⎧ ! D9 cos θ̂
⎫
⎨ (Im{[b] }2 +Im{[b] }2 )2 +(Re{[b] }Im{[b] }+Re{[b] }Im{[b] })2 ⎬
1
2
1
1
2
2
γ̂ = tan−1
|
Re
{[b]
}
Im
{[b]
}−
Re
{[b]
}
Im
{[b]
}|
⎩
⎭
1
2
2
1
⎫
⎧ ⎧
⎪
⎬
⎨ (Re{[b] }Im{[b] } + Re{[b] }Im{[b] }) ⎪
⎪ ⎪
⎪
1
1
2
2
⎪
⎪
,
if sin η ≥ 0,
⎪
⎪
⎪
⎪
⎭
⎩ + j (Im{[b] }2 + Im{[b] }2 )
⎨ ⎪
1
2
⎫
⎧
η̂ =
⎪
⎪
⎪
⎪
⎪ ⎨ (Re{[b]1 }Im{[b]1 } + Re{[b]2 }Im{[b]2 }) ⎬
⎪
⎪
⎪
+ π, if sin η < 0.
⎪
⎪ ⎪
⎪
⎩
⎭
⎩ + j (Im{[b] }2 + Im{[b] }2 )
1
2
(Im{α})(tan φ) × (4) = (Re{α})(cos θ ) × (9)
−(Im{α})(cos θ ) × (6)
+(Re{α})(tan φ) × (7).
V. C ONCLUSION
Azimuth-elevation direction finding and polarization estimation are investigated for all 20 possible different compositions
of a collocated triad of orthogonally oriented dipole(s) and/or
loop(s). For the 4 compositions without any dipole and any
loop oriented along the z-axis, closed-form estimation is not
viable. Closed-form estimation-formulas are produced for
16 compositions, 14 of these were previously unavailable in
the open literature. The dipole-triad and the loop-triad alone
(among all 20 compositions) allow unambiguous directionof-arrival estimation over the entire sphere. The other 14
compositions have azimuth-elevation arrival-angles estimates
with only an hemispherical (not spherical) validity region, due
to the hemispherical ambiguity in the trigonometric functions.
A PPENDIX : T HE D ETAILED D ERIVATION FOR
C OMPOSITION 1.3: {ex , e y , h z }
To demonstrate detailed algebraic and trigonometric manipulations leading to the estimation-formulas in Tables I-V,
Prior Info.
Required
def
c = âe− j [â]2
D7 =
D8 =
Im{[c]1 } sin γ̂
Re{[c]2 } cos γ̂ sin η̂
−Im{[c]3 } sin γ̂
Re{[c]2 } cos γ̂ sin η̂
cos φ
= sin θ
sin φ cos θ
=
sin θ
θ ∈ [0, π2 ]
or θ ∈ ( π , π ]
2
η ∈ [0, π )
or
def
b = âe− j [â]3
Im{[b]2 } cos γ̂
cos φ
= sin θ
Re{[b]3 } sin γ̂ sin η̂
Im{[b]1 } cos γ̂
sin φ cos θ
=
=
sin θ
Re{[b]3 } sin γ̂ sin η̂
η ∈ [−π, 0)
θ ∈ [0, π2 ]
or θ ∈ ( π , π ]
2
D9 =
D10
η ∈ [0, π )
or
η ∈ [−π, 0)
this appendix will use Composition 1.3: {ex , e y , h z } as an
illustrative example.
The derivation here will start from the left-hand side of
⎤
⎡ ⎤
⎡
ex
cos φ cos θ sin γ e j η − sin φ cos γ
â = α ⎣ e y ⎦= α ⎣sin φ cos θ sin γ e j η + cos φ cos γ⎦ (10)
hz
sin θ cos γ
where α represents an unknown complex-value number that
may have arisen from eigen-based data-processing as discussed in footnote 3.
As θ ∈ [0, π] and γ ∈ [0, π/2), it is true that sin θ
cos γ ≥ 0. Hence, define
[â]3
b = âe− j
⎡
def
⎤
cos φ cos θ sin γ cos η − sin φ cos γ
= |α| ⎣ sin φ cos θ sin γ cos η + cos φ cos γ ⎦
sin θ cos γ
⎡
⎤
cos φ cos θ sin γ sin η
+ j |α| ⎣ sin φ cos θ sin γ sin η ⎦
0
where
(11)
denotes the angle of the ensuing entity. From (11),
Re{[b]1 } = |α|(cos φ cos θ sin γ cos η − sin φ cos γ ),
(12)
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IEEE SENSORS JOURNAL, VOL. 12, NO. 6, JUNE 2012
Re{[b]2 } = |α|(sin φ cos θ sin γ cos η + cos φ cos γ ),
(13)
Re{[b]3 } = |α|(sin θ cos γ ),
(14)
Im{[b]1 } = |α|(cos φ cos θ sin γ sin η),
Im{[b]2 } = |α|(sin φ cos θ sin γ sin η).
(15)
(16)
A. To Derive φ̂
From (15)-(16),
Im{[b]2 }
.
(17)
tan φ =
Im{[b]1}
As tan φ = tan(φ + π),
⎧
(
)
⎨tan−1 Im{[b]2 } ,
if (cos θ sin η)Im{[b]1 } ≥ 0
( Im{[b]1 } )
φ̂ =
{[b]
}
Im
2
⎩tan−1
Im{[b]1 } + π, if (cos θ sin η)Im{[b]1 } < 0.
(18)
The above conditions arise from the following consideration:
(cos θ sin η)Im{[b]1} ≥ 0 ⇒ cos φ ≥ 0 ⇒ φ ∈ [−π/2, π/2]
(cos θ sin η)Im{[b]1 } < 0 ⇒ cos φ < 0 ⇒ φ ∈ (π/2, 3π/2),
which help to determine whether φ ∈ [−π/2, π/2] or φ ∈
(π/2, 3π/2). The inequalities in (18) require prior knowledge
which of cases (i ) or (ii ) holds:
(i ) θ
θ
(ii ) θ
θ
∈ [0, π/2] ∩ η ∈ [−π, 0)
∈ (π/2, π] ∩ η ∈ [0, π)
∈ [0, π/2] ∩ η ∈ [0, π)
∈ (π/2, π] ∩ η ∈ [−π, 0)
⇒
⇒
⇒
⇒
cos θ sin η ≤ 0,
cos θ sin η ≤ 0,
cos θ sin η ≥ 0,
cos θ sin η ≥ 0.
B. To Derive θ̂
From (12)-(13),
Re{[b]2 } cos φ̂ − Re{[b]1 } sin φ̂ = |α| cos γ .
(19)
Together with (14),
Re {[b]3 }
.
Re {[b]2 } cos φ̂ − Re {[b]1 } sin φ̂
As sin θ = sin(π − θ ) for θ ∈ [0, π],
⎧
Re{[b]3 }
⎪
sin−1 ⎪
,
⎪
{[b]2 } cos φ̂−Re{[b]1 } sin φ̂
Re
⎪
⎪
⎨
if θ ∈
[0, π/2];
θ̂ =
Re{[b]3 }
−1
⎪
π − sin
⎪
Re{[b] } cos φ̂−Re
,
⎪
{[b]1 } sin φ̂
⎪
2
⎪
⎩
if θ ∈ (π/2, π].
sin θ =
(20)
(21)
The above requires prior knowledge of whether θ ∈ [0, π/2]
holds or θ ∈ (π/2, π] holds.
C. To Derive η̂
From (12)-(13),
[b]1 cos φ̂ + [b]2 sin φ̂ = |α| cos θ sin γ cos η
+ j |α| cos θ sin γ sin η
= |α| cos θ sin γ (cos η + j sin η)
= |α| cos θ sin γ e j η .
As |α| sin γ ≥ 0,
η̂ =
*
[b]1 cos φ̂ + [b]2 sin φ̂
cos θ̂
(22)
+
.
(23)
D. To Derive γ̂
From (14) and (15),
Im{[b]1 }
= tan γ (cos φ cot θ sin η),
Re{[b]3 }
Im{[b]1 } tan θ
tan γ =
.
Re{[b]3 } cos φ sin η
(24)
As sin γ ≥ 0,
+
*
Im{[b] } tan θ̂ 1
γ̂ = tan−1 .
Re{[b]3} cos φ̂ sin η̂ (25)
E. Validity Region
Examining the prior knowledge required by the
four estimation-formulas of (18), (21), (23) and
(25), those estimation-formulas’ validity region equals
{θ ∈ [0, π2 ] or θ ∈ ( π2 , π]} ∩ {φ ∈ [0, 2π)} ∩ {γ ∈
[0, π2 )} ∩ {η ∈ [−π, 0) or η ∈ [0, π)}.
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Kainam Thomas Wong (SM’01) received the
B.S.E. degree in chemical engineering from the
University of California, Los Angeles, the B.S.E.E.
degree from the University of Colorado, Boulder,
the M.S.E.E. degree from Michigan State University,
East Lansing, and the Ph.D. degree in electrical
and computer engineering from Purdue University,
West Lafayette, IN, in 1985, 1987, 1990, and 1996,
respectively.
He was a Manufacturing Engineer with General
Motors Technical Center, Warren, MI, from 1990 to
1991, and a Senior Professional Staff Member with Johns Hopkins University
Applied Physics Laboratory, Laurel, MD, from 1996 to 1998. From 1998 to
2006, he had been a Faculty Member with Nanyang Technological University,
Singapore, the Chinese University of Hong Kong, Hong Kong, and the
University of Waterloo, Waterloo, ON, Canada. Since 2006, he has been with
Hong Kong Polytechnic University, Kowloon, Hong Kong, as an Associate
Professor. His current research interests include sensor-array signal processing
and signal processing for communications.
Dr. Wong has been an Associate Editor for the IEEE S IGNAL P ROCESSING
L ETTERS from 2006 to 2010 and Circuits, Systems, and Signal Processing from 2007 to 2009. He has been an Associate Editor for the IEEE
T RANSACTIONS ON V EHICULAR T ECHNOLOGY since 2007 and the IEEE
T RANSACTIONS ON S IGNAL P ROCESSING since 2008. He was conferred the
Premier’s Research Excellence Award in 2003 by the Canadian province of
Ontario.
Xin Yuan (S’09) received the B.Eng. degree in
electronic information engineering and the M.Eng.
degree in information and communication engineering from Xidian University, Xi’an, China, in 2007
and 2009, respectively. He is currently pursuing the
Ph.D. degree at Hong Kong Polytechnic University,
Kowloon, Hong Kong.
His current research interests include diversely
polarized antenna-array signal processing.
Keshav Agrawal received the B.Tech. degree in
electrical engineering from the Indian Institute of
Technology, Kanpur, India, in 2011. He is currently
pursuing the M.S.E.E. degree at the University of
California, Los Angeles.
He was a Research Assistant with Hong Kong
Polytechnic University, Kowloon, Hong Kong, in
June and July of 2011. His current research interests include sensor networks and sensor-array signal
processing.
Zixin Xu received the B.Eng. degree in communications engineering from the University of Electronic
Science and Technology of China, Chengdu, China,
and the M.Sc. degree in electronic and information
engineering from Hong Kong Polytechnic University, Kowloon, Hong Kong, in 2005 and 2011,
respectively.
He was a Software Engineer and a Project Manager in Shanghai from 2005 to 2009. Since 2011,
he has been a Software Engineer in Ericsson Radio
Technology Co. Ltd. in Chengdu.