社団法人 電子情報通信学会
THE INSTITUTE OF ELECTRONICS,
INFORMATION AND COMMUNICATION ENGINEERS
信学技報
TECHNICAL REPORT OF IEICE.
球波動関数を用いた双角度チャネルのモデル化
中井 健二†
高田 潤一†
ダシティ マルジィエ†
金 ミンソク†
トミー ライティネン††
† 東京工業大学 大学院理工学研究科 国際開発工学専攻 東京 日本
†† ヘルシンキ工科大学 無線科学工学専攻 ヘルシンキ フィンランド
E-mail: †{nakai,takada,mskim,dashti}@ap.ide.titech.ac.jp, ††[email protected]
あらまし 無線チャネルを測定・解析・モデル化する際に, 通信機器の一部である送受信アンテナをチャネルから切り
出し, アンテナに依存しない電波伝搬路モデルを構築するアプローチは双角度チャネルモデルと呼ばれている. 本稿で
は, 双角度チャネルモデルを球波動関数を用いて再構築することを目的とし, 実際のアンテナ及び伝搬路特性に対して
球波動関数による表現を試みる. さらに, 球波動関数を用いて再構成した複素放射パタンを示し, また再構成のために
必要とされるモード数を議論する.
キーワード
双角度チャネルモデル, 球面波, 球面調和関数展開, 打ち切り
Double-directional channel with the spherical harmonics
Kenji NAKAI† , Jun-ichi TAKADA† , Minseok KIM† ,
Dashti MARZIEH† , and Tommi LAITINEN††
† Graduate School of Engineering, Tokyo Institute of Technolog , Tokyo, Japan
†† Department of Radio Science and Engineering, Aalto University School of Science and Technology,
Helsinki, Finland
E-mail: †{nakai,takada,mskim,dashti}@ap.ide.titech.ac.jp, ††[email protected]
Abstract There is an approach that extracts or de-embeds the transmitting and receiving antennas from the
multipath radio channels, as they are parts of communication devices of the channel in particular in the state-of-art
MIMO technology. This approach is called double-directional channel, since it constructs a radio propagation model
independent of antennas. This paper presents to construct double-directional channel with spherical harmonics to
try to express antennas and propagation properties for the preparation of detailed data analysis, the convergence
property of the antenna pattern reconstruction by truncated numbers of the spherical harmonics.
Key words Double-directional channel modeling, Spherical waves, Spherical wave expansion, Truncation
1. Introduction
Conventionally, the double directional channel is expressed
as superposition of local plane waves at both transmitter and
There is an approach that extracts or de-embeds the trans-
receiver antennas. Although the radio wave emitted from
mitting and receiving antennas from the multipath radio
antenna propagates as a spherical wave, Rx antenna is so
channels, as they are parts of communication devices of the
far from Tx antenna that this spherical wave is modeled as
channel in particular in the state-of-art MIMO technology.
a plane wave locally. Moreover, in the scattering processes
This approach is called double-directional channel, since it
such as reflection and diffraction, image source and diffrac-
constructs a radio propagation model independent of anten-
tion point are also very far from both Tx and Rx, so that
nas. The approach enables the comparison of performances
they are also modeled as plane waves.
of different antennas in the same propagation environment
However, a scattering phenomenon of actual radio wave is
by embedding the antennas into the double-directional chan-
represented by the radiation from the distributed equivalent
nel.
source on the scatterer. When the scatterer is very close to
—1—
K 1mn (θ, ϕ) = √
(
(
−
m
|m|
)m
exp[imϕ](−i)n+1
)
|m|
|m|
imP¯n (cos θ) ˆ dP¯n (cos θ) ˆ
θ−
ϕ ,
sin θ
dθ
K 2mn (θ, ϕ) = √
(
n
(−i)
Fig. 1 Double-directional channel expressed by sherical wave
2
2n(n + 1)
2
2n(n + 1)
(
−
m
|m|
(1)
)m
exp[imϕ]
)
|m|
|m|
dP¯n (cos θ) ˆ imP¯n (cos θ) ˆ
θ+
ϕ ,
dθ
sin θ
(2)
where n = 1, 2, 3, ... and m = −n, −n + 1, ..., 0, ..., n − 1, n,
|m|
and P¯n (cos θ) is the normalized associated Legendre function. In general, n and m represents the mode indices in θ
either Tx or Rx antennas or the surface of the scatterer is
so complicated, it is difficult to represent the scattered wave
as a superposition of plane waves. To resolve these plane
and ϕ.
By using spherical harmonics, arbitrary radiation pattern
from the source in angular domain can be expressed as
waves, the array antennas with very large aperture size is
necessary, although the size of the array antenna is usually
limitted by the cost and portability. This is owing to the fact
that the plane waves are represented by continuous angular
spectrum. Alternatively, the radiated field of the antennas
are expressed as the superposition of the spherical harmonics. Due to the periodic nature to the angle, the spherical
harmonics are discrete. Moreover, the upper bound of the
size of the antenna [1]. This approach of decomposition of
antenna pattern into spherical harmonics is very popular in
spherical near-field measurement of the antenna [2].
This paper tries to express the double-directional channel
by use of vector spherical harmonics. Not only the antennas, but also the propagation channel are expressed by using
spherical harmonics as shown in Fig.1. The advantage of the
use of spherical harmonics is that the resolution in angular
domain is automatically determined by the size of the antenna. In contrast, the required angular resolution of the
plane wave is infinite regardless of the antenna size.
This paper reviews the basic spherical wave theory [2 − 5]
at first. Next, how to find the unknown coefficients for the
spherical harmonics expansion of the antenna pattern is explained.
By utilizing the measured antenna pattern, the
convergence property of the finite sum of the spherical harmonics. A new modeling approach of the double-directional
channel using the spherical harmonics is finally proposed.
2. Spherical Wave Theory
In this paper, time dependence of exp(−iωt) is assumed.
By solving the free space homogenious vector Helmholtz
equation in spherical coordinates, vector spherical harmonics
are obtained as the solutions. Their angular components are
expresed as
2
N
n
k ∑∑ ∑
E(θ, ϕ) = √
Qsmn K smn (θ, ϕ),
η s=1 n=1 m=−n
(3)
√
where η is characteristic impedance given by η = (ϵ/µ), k
√
is the propagation constant given by k = ω µϵ = 2π
, λ is
λ
the wavelength. With respect to the index s, s = 1 indicates
TE-wave, while s = 2 indicates TM-wave.
Generally, an infinite number of spherical wave modes
would be required to exactly characterise the radiated field
of a general source. However, the spherical wave series can
be truncated at some certain n = N i.e. a sufficient value of
N for convergence is determined by the size of the antenna.
3. Experimetal Investigation of Antenna
Pattern Representation by using Spherical Harmonics
3. 1 Antenna Under Test and Measurement Setup
The antenna pattern measurement has been conducted by
using SATIMO Stargate, which is a spherical near field measurement system. The measurement is conducted for whole
sphere to get the radiation pattern in whole solid angle. A
biconical antenna in Fig.2 was measured at 2.385 GHz. Obviously this measurement system itself may use the same
spherical harmonics approach inside the system, it just outputs far field vector antenna pattern in whole sphere.
3. 2 Measured Radiation Pattern
Fig.3 shows the linear amplitude of Eθ -component, while
Fig.4 shows Eϕ -component. Since the antenna is vertically
polarized, Eϕ is very small compared with Eθ .
3. 3 Pattern Decomposition into the Spherical
Harmonics
From the measured radiation pattern, the coefficients Qs
—2—
Fig. 2 Coordinate system about the biconical antenna
in Eq.(3) can be derived by solution the following set of linear
Fig. 3 Complex radiation pattern of Eθ
equations.
E = KQ,
(4)
Eθ (θ1 , ϕ1 )
..
.
Eθ (θR , ϕ1 )
..
.
)
(
Eθ (θR , ϕS )
Eθ
,
=
E=
Eϕ
Eϕ (θ1 , ϕ1 )
..
.
Eθ (θR , ϕ1 )
..
.
Eϕ (θR , ϕS )
(
K=
Kθ
(5)
Fig. 4 Complex radiation pattern of Eϕ
least square solution, i.e.
Q = K − E,
where K − is Moore-Penrose pseudo inverse of K
)
3. 4 Pattern Reconstruction using Finite Spherical Harmonics
Kϕ
Kθ,1,1(−1)1 . . . Kθ,1,1N N Kθ,1,2(−1)1 . . . Kθ,1,2N N
..............................
Kθ,R,1(−1)1 . . . Kθ,R,1N N Kθ,1,2(−1)1 . . . Kθ,R,2N N
=
Kϕ,1,1(−1)1 . . . Kϕ,1,1N N Kθ,K,2(−1)1 . . . Kθ,1,2N N
..............................
By submiting Eq.(8) into Eq.(3), vector radiation pattern
is reconstructed i.e. unknown coefficient Q is solved simulta-
Fϕ,S,1(−1)1 . . . Kϕ,S,1N N Fθ,S,2(−1)1 . . . Kθ,S,2N N
(6)
(8)
neous equation about it. Now, the question is how to select
appropriate N . Here, relative error ϵ is considered as a function of N , which is defined as
)
∑ ∑ (
2
2
sin θS
R
S |△Eθ (θR , ϕS )| + |△Eϕ (θR , ϕS )|
)
ϵ= ∑ ∑ (
,
2 + |E (θ , ϕ )|2 sin θ
|E
(θ
,
ϕ
)|
R
S
S
θ R
ϕ R
R
S
(9)
Q1(−1)1
..
.
Q1N N
,
Q=
Q2(−1)1
..
.
Q2N N
where, the error at individual angle, sin θ is weighted considering the density of the point and △ means the difference of
(7)
reconstructed the electric field and measured the field.
Fig.5 shows the relative error as function of N for the
measured antenna. For example, to suppress the error to 3%,
N should be at least 3. Fig.6 compares the measured and
where E is measured the electric field vector, K is matrix of
the reconstructed pattern for N = 3. Although the distor-
the spherical wave functions and Q is weighted coefficients
tion of the pattern in Fig.7 is still obvious, the whole spheres
vector. Each of index (R, S) is the numbers of samples in ele-
shows the reconstructed pattern with N = 11. It is quite
vation angle and azimuth angle. The others indices (s, m, n)
more similar to the measured pattern than that with N = 3.
are the same as Eq.(3).
Usually, Eq.(4) is overdetermined and Q is derived as the
—3—
Fig. 8 Equivalent circuit model
y(t) consists of Qs of Tx antenna output and received antenna input. Therefore,
Fig. 5 Relative error vs N for measured antenna
y(t) = H prop x(t)
(10)
where H prop is propagation channel matrix determined for
4. Double-directional Channel Model using Spherical Harmonics
spherical harmonics. The authors target the chanel model
expressed by spherical harmonics in the next step. In the
similar manner, Tx antenna and Rx antenna are also ex-
By representing antennas using Eq.(3), the double-
pressed as
directional channel can also be expressed as an equivalent
circuit of multiport circuit. Multipath environment such as
x(t) = H T x vin (t),
Fig.1 can be represented as Fig.8, where each individual port
vout (t) = H Rx y(t).
corresponds to a mode of spherical harmonics.
Transmitting signal vector x(t) for received signal vector
(11)
Here, it is already defined in Eq.(3) that H T x ∝ QT x and
its reciprocal at Rx as H Rx ∝ Q∗Rx , NT x and NRx are determined by the size of the antenna. By using spherical harmonics, equivalent circuit of the channel can be derived.
5. Conclusions and Future Works
.
This paper presented the double-directional channel
modeling frame work. By using measured antenna pattern,
the antenna model can be derived. In the next step, the
double-directional propagation channel will be modeled in
the same manner. Some simple antennas and Los channel
may be a good starting point to validate the proposed approach.
Fig. 6 Comparison between measured pattern and reconstructed
Acknowledgments
pattern with N = 3
The authors thank Dr. Ichirou Ida and Mr. Atsushi Honda
of FUJITSU LABORATORIES LIMITED for their cooperation in the antenna measurement, and Dr. Katsuyuki
Haneda of Aalto University School of Science and Technology for his advice. This project is supported by JSPS and
AF under the Japan - Finland Bilateral Core Program.
References
Fig. 7 Comparison between measured pattern and reconstructed
pattern with N = 11
[1] K. Araki, T. Ohira and J. Takada, ”Eigen mode expansion
of free space and its application to an expression for antenna
equivalent circuit”, IEICE Tech. Rep, A.P2006-128, Jan.
[2] J.E.Hansen, Spherical Near-Field Antenna Measurement,
London, U.K, pp.1-60,1988
[3] C.T. Tai, Dyadic Green Functions in Electromagnetic Theory 2nd ed, IEEE press, pp. 198-224, ,1993
[4] J.A.Stratton, Electromagnetic Theory, New York, McGrawHill Book Company, pp. 392-412, 1941
[5] T.Laitinen, ”Spherical wave expansion based measurement
—4—
procedures for radiated fields”, Doctor dissertation, Helsinki
University of Technology, Finland 2000.
—5—
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