2009 International ITG Workshop on Smart Antennas – WSA 2009, February 16–18, Berlin, Germany
ARRAYS OF ISOTROPIC RADIATORS — A FIELD-THEORETIC JUSTIFICATION
Hristomir Yordanov, Peter Russer
Michel T. Ivrlač, Josef A. Nossek
Institute for High Frequency Engineering
Technische Universität München
yordanov@tum.de
Institute for Circuit Theory and Signal Processing
Technische Universität München
ivrlac@tum.de
ABSTRACT
Because of their conceptual simplicity and mathematical convenience, isotropic radiators are all too popular in the analytical treatment of antenna arrays. However, because an isotropic
electro-magnetic field is physically impossible, it is legitimate
to ask on which ground the assumption of having isotropic radiators in the theoretical analysis of radio communications is
justified. In this paper, we provide a field-theoretic answer to
this question. To this end, we show that a uniform linear array
(ula) of isotropic radiators leads qualitatively to the same relative antenna coupling as if the ula was composed of Hertzian
dipoles. This implies that the transmit array gain – that is, the
reduction of necessary transmit power to generate the same
electric field strength at a given point of interest, compared to
using a single antenna of the array – of a ula of isotropic radiators, is qualitatively the same as of a ula of Hertzian dipoles.
This shows that, albeit isotropic antennas do not exist, their
application in theoretical investigation of antenna arrays is legitimate, because they lead to physically reasonable antenna
coupling which allows a qualitatively correct analysis of important performance parameters, like array gain.
1. INTRODUCTION
Antenna arrays can greatly enhance the performance of radio
communication systems by increasing link reliability, capacity
and coverage [1]. Therefore, antenna arrays are the object of
study in a variety of different disciplines, like electro-magnetic
field theory, signal and system theory, coding theory, and information theory. Of course, the methods and notions used are
different in each discipline. In the signal processing and information theory literature, it is common to exercise a rather simplistic view on antenna arrays: the individual antennas of the
antenna array are usually assumed to be isotropic radiators (see,
e.g. [2], chapter 7). However, the appealing conceptual simplicity and the mathematical convenience of such an approach are
overshadowed by the very fact that antennas which excite an
isotropic vector-field do not exist [3]. Therefore, it is an interesting question, on which ground one can justify isotropic radiators in the analytical treatment of antenna arrays.
In this paper, we provide a field-theoretic justification of the
legitimacy of the use of isotropic radiators in the theoretical
treatment of antenna arrays. We show that a uniform linear array (ula) of isotropic radiators leads to qualitatively the same
relative antenna mutual coupling as if the ula was composed
of Hertzian dipoles. Because the transmit power is directly related with antenna coupling [4], it follows that the transmit array gain1 of a ula composed of isotropic radiators is qualitatively the same as the transmit array gain of a ula of Hertzian
dipoles. Albeit isotropic antennas do not exist, their application in theory is legitimate, because they lead to an antenna
coupling, which is physically reasonable. This allows qualitatively correct analysis of important performance parameters,
like array gain, without having to give up simplicity and convenience which isotropic radiators offer in system modeling.
2. UNIFORM LINEAR ARRAY OF ISOTROPS
A ula of N isotropic radiators can be seen as a linear N-port.
Assuming lossless radiators, the radiated power is equal to the
electric input power:
PTx = Re{V H I} ,
(1)
PTx = I H Re{Z}I.
(2)
where V ∈ C N×1 ⋅ V, and I ∈ C N×1 ⋅ A, are the complex voltage
and current envelopes, respectively, while Re{.}, and (.)H are
the real-part, and the Hermitian operations, respectively. Because of linearity, we have that V = ZI, where Z ∈ C N×N ⋅ Ω
is the impedance matrix of the array. Because of reciprocity,
there is Z = Z T , and hence, the transmit power can be written:
Let us now compute Re{Z}. The strength of the electric farfield in a distance r and elevation angle θ, excited by the ula
located in the origin and aligned with the z-axis, can be written
as (see e.g., [3] on pp 250 and 258):
E=α
e−j kr H
a I.
r
(3)
1The transmit array gain is the factor by which the transmit power can be
reduced compared to a single antenna of the array, while the same strength of
the electric field is excited at a position of interest.
2009 International ITG Workshop on Smart Antennas – WSA 2009, February 16–18, Berlin, Germany
Herein, a = ( 1 e jν ⋯ e j(N−1)ν ) , with ν = kd cos θ, where d
is the distance between the radiators, k = 2π/λ, with λ denoting the wavelength, and α is a constant. The power density is
then given by (see e.g., [5], eq. (8) on page 571):
z
T
α ′ ∣α∣ H H
S = α ∣E∣ =
I aa I,
r2
′
2
2 H
H
sin(θ) dθ) I. (5)
0
sphere
Since PTx = I H Re{Z}I = γ ⋅ I H C I, it follows from (5) that
(Re{Z})m,n = γ ⋅ (C)m,n = γ ⋅ j 0 (kd∣m − n∣),
(6)
where γ = 4πα ′ ∣α∣ , is a constant, and
2
sin(x)
.
j 0 (x) =
x
(7)
Notice that the mutual coupling matrix C ∈ C N×N, is a dimensionless, real-valued Toeplitz matrix, which has got unity entries all along its main diagonal:
⎡ 1
j 0 (kd) j 0 (2kd) j 0 (3kd) ⋯ ⎤
⎢
⎥
⎢
⎥
⎥.
j
(kd)
1
j
(kd)
j
(2kd)
⋱
C=⎢
0
0
⎢ 0
⎥
⎢
⎥
⎢ ⋮
⎥
⋱
⋱
⋱
⋱
⎣
⎦
(8)
The real-part of the impedance matrix can therefor be written
compactly as:
Re{Z} = γ ⋅ C.
(9)
It is interesting to note that the computation of the mutual coupling matrix for the ula of lossless isotropic radiators, is based
solely on the superposition principle for the electric field, as
expressed in (3), and the law of conservation of energy, from
which (2) follows. Only for specific antenna separation the coupling matrix is the identity, namely only for kd/π = 2d/λ ∈ N.
For all other antenna separations, the very law of conservation
of energy enforces that there is mutual coupling!
3. UNIFORM LINEAR ARRAY OF HERTZIAN DIPOLES
Let us first consider an array of two collinear Hertzian dipoles,
located at a distance d from each other and aligned with the
z axis of a rectangular coordinate system. The dipoles are situated on the x axis, as shown in Figure 1. We can consider the
system of two dipoles as a two-port, which can be described
by its impedance matrix. In particular:
V 1 = Z 1,1 I 1 + Z 1,2 I 2
,
V 2 = Z 2,1 I 1 + Z 2,2 I 2
2
x
(4)
π
∫∫ S dA = 2πα ∣α∣ I ( ∫ aa
′
1
d
where α ′ ∈ R+ ⋅ Ω−1 is a constant. We can obtain the transmit
power by integrating the power density over a sphere around
the array:
PTx =
h
2
(10)
Fig. 1. An array of two Hertzian dipoles
where I 1 , and I 2 are the current phasors which are impressed
on antenna 1 and 2, respectively, while V 1 , and V 2 are the corresponding induced voltage phasors. The Z 11 and the Z 22 elements of this matrix are defined as
Z 1,1 =
V1
∣
I1 I
Z 2,2 =
,
2 =0
V2
∣
.
I 2 I =0
(11)
1
The conditions I 2 = 0 and I 1 = 0 mean that the matrix element
Z i ,i is simply the input impedance of the i-th antenna. The real
part of the input impedance of a Hertzian dipole is given by [4]
h 2
2
(12)
R r = Re{Z 1,1 } = Re{Z 2,2 } = πZ F0 ( ) ,
3
λ
where Z F0 ≈ 377 Ω, is the free-space wave impedance, and λ
is the free-space length of the radiated wave. Because of reciprocity, we have Z 1,2 = Z 2,1 . As of (10), the mutual impedance
Z 2,1 is defined as
V
Z 2,1 = 2 ∣
,
(13)
I 1 I =0
2
that is, as the ratio between the induced voltage on the second
antenna due to the impressed current on the first antenna, provided that the current on the second antenna vanishes. We can
easily compute the induced voltage on the second antenna by
integrating the tangential component of the electric field of the
first antenna over the length of the second antenna [6]. The
electric field of a Hertzian dipole, located in the origin of the
coordinate system due to an impressed current I 1 is given in
spherical coordinates as:
Eθ =
hI 1
k 2 jk
1
(− + 2 + 3 ) e−jkr sin θ,
4πjωε 0
r
r
r
(14)
Er =
hI 1
jk
1
( + ) e−jkr cos θ,
(14a)
2πjωε 0 r 2 r 3
√
where k = 2π/λ = ω ε 0 µ 0 , and
√
r = x 2 + y2 + z2 ,
(15)
√
x 2 + y2
(15a)
θ = arctan
z
The voltage which can be measured at the terminals of the receiving dipole antenna is given by the following expression:
V = −
∫
E⃗ ⋅ d⃗l ,
Antenna
(16)
2009 International ITG Workshop on Smart Antennas – WSA 2009, February 16–18, Berlin, Germany
where E⃗ is the electric field which is excited by the antenna,
and the integration is performed over the antenna length. For
the configuration, shown in Figure 1, we have:
E⃗ ⋅ d⃗l = E z dz.
j 0 (x)
0.8
(17)
2
hI 1
k
jk
1
(− + 2 + 3 ) e−jkd . (18)
4πjωε 0
d d
d
j 0 (x), Ψ(x)
0.6
Since the Hertzian dipole is electrically small, i.e. kh ≪ 1, we
can approximate the z-component of the electric field on the
antenna by the θ component
E z ≈ E θ (θ = π/2) =
1
Ψ(x)
0.4
0.2
0
−0.2
−0.4
0
0.5
The voltage between the terminals of the second antenna is:
h/2
−h/2
V2 = −
h/2
2
θ
θ
−h/2
2
h I1
k
jk
1
(− + 2 + 3 ) e−jkd .
=
4πjωε 0
d d
d
(19)
Therefore, the mutual impedance is given by
V2
k 2 jk
1
h2
(− + 2 + 3 ) e−jkd .
(20)
=
I1
4πjωε 0
d d
d
√
√
Using k = 2π/λ = ω ε 0 µ 0 , and Z F0 = µ 0 /ε 0 , as well as (12),
the real part of this expression can be rewritten as
Z 2,1 =
Re{Z 2,1 } = R r ⋅ Ψ(kd),
1.5
2
2.5
3
x/(2π)
Fig. 2. The functions j 0 (x) from (7), and Ψ(x) from (22).
∫ E dz = ∫ E dz = E h =
θ
1
j 0 (x) and Ψ(x) appear to be rather different. However, when
we look at Figure 2, we observe that there actually is a striking similarity. Both functions are decreasing at first, when x
is increased from zero, and then display an oscillatory behavior, as x is further increased. While j 0 (x) has equidistant roots
(integer multiples of π), the function Ψ has almost equidistant
roots at almost the same positions as j 0 (x). From what we see
in Figure 2, we may therefore conclude, that j 0 (x), and Ψ(x)
are qualitatively the same. This justifies the use of isotropic radiators in the theoretical treatment of antenna arrays, because
they impose qualitatively the same relative mutual coupling.
(21)
4. TRANSMIT ARRAY GAIN
where
3 sin x cos x sin x
Ψ(x) = (
+ 2 − 3 ).
(22)
2
x
x
x
This result can easily be generalized to the case of more than
two Hertzian dipoles. For a ula of N Hertzian dipoles, we obtain the following real-part of the impedance matrix:
Re{Z H } = R r ⋅ C H ,
(23)
Array Gain is an important performance measure of antenna
arrays. Because array gain critically depends on antenna mutual coupling, we expect – from what was said before – to find
qualitatively the same results no matter if the array is equipped
by Hertzian dipoles, or isotropic radiators. Let us check this.
From (3) and (14), the electric far-field strength excited by a
ula of Hertzian dipoles can be written as:
E = ξ ⋅ a H I ⋅ sin θ,
where
⎡ 1
Ψ(kd) Ψ(2kd) Ψ(3kd) ⋯ ⎤
⎢
⎥
⎢
⎥
⎢
1
Ψ(kd) Ψ(2kd) ⋱ ⎥
C H = ⎢ Ψ(kd)
⎥.
⎢
⎥
⎢ ⋮
⎥
⋱
⋱
⋱
⋱
⎣
⎦
(24)
What is the major difference of the real-part of the impedance
matrix of the ula of Hertzian dipoles, when we compare it
to the case of isotropic radiators? To answer this question, we
just have to look at the pair of equations (24) and (23) for the
case of Hertzian dipoles, and compare it with the corresponding pair of equations (8) and (9) for the case of isotropic radiators. Obviously, the main difference is that the relative mutual
coupling is described by the function j 0 (x) from (7) for the
case of isotropic radiators, while it is described by the function
Ψ(x) from (22), for the case of Hertzian dipoles. At first glance,
(25)
where ξ is a (distance-dependent) constant, and a ∈ C N×1 is
the (direction dependent) Vandermonde array steering vector.
Hence,
2
2
(26)
∣E∣ = ∣ξ∣ ⋅ I H aa H I ⋅ sin2 θ.
The power PRx , that can be extracted from the field by an antenna is then proportional to (26), such that:
PRx = ζ ⋅ I Haa H I,
(27)
where ζ ∈ R+ ⋅ Ω is proportional to ∣ξ∣2 sin2 θ. The optimum
excitation current, for a given steering vector a, maximizes the
received signal power for a given transmit power PTx :
I opt = arg max I H aa H I, s.t. R r ⋅ I HC H I ≤ PTx .
I
(28)
2009 International ITG Workshop on Smart Antennas – WSA 2009, February 16–18, Berlin, Germany
max
PRx
=
PTx ζ H −1
a C H a.
Rr
(29)
If only one transmit antenna, say the n-th one, is excited by a
non-zero electric current, the receive power becomes:
PRx,n =
PTx ζ
2
∣a n ∣ ,
Rr
where a n is the n-th component of the steering vector a. The
average received power over all transmit antennas becomes
PRx, avr =
single
(30)
such that the transmit array gain is defined as:
A Tx =
max
PRx
single
PRx, avr
.
A Tx
N=4
(»end-fire«)
N =3
N =2
15
10
5
0
0
0.5
1
1.5
2
2.5
3
Fig. 3. Transmit array gain in »end-fire« direction, as function
of antenna spacing for different antenna numbers. Results for
Hertzian dipoles and isotropic radiators are shown.
(31)
5. CONCLUSION
When we substitute (29), and (30) into (31), it follows that
−1
a
a HC H
= N
.
H
a a
20
Isotropic radiators
Hertzian dipoles
Antenna spacing, d/λ
N
PTx ζ
2
∑ ∣a n ∣ ,
N R r n=1
N=5
25
Transmit Array Gain, A Tx
The transmit power constraint follows from substituting (23)
for Re{Z} in (2). It can be shown that the receive power under
optimum beamforming is given by [7]:
(32)
Note that A Tx does not depend on R r , or ζ, but solely on the
array steering vector, and the relative antenna mutual coupling
matrix. We can obtain the transmit array gain for the case of a
ula of isotropic radiators by simply replacing in (32), the matrix C H from (24), by the matrix C from (8). Notice that A Tx
quantifies how much more receive power we obtain by optimally using all the antennas of the array, compared to a single
transmit antenna of the same type as used in the array, when
the radiated power is the same in both cases.
For both a ula of Hertzian dipoles and isotropic radiators,
Figure 3 shows the array gain in the direction of the array axis,
the so-called »end-fire« direction, for different antenna number N, as a function of the antenna spacing. We can observe
that the array gain raises quickly as we reduce the antenna spacing below λ/2. The maximum array gain happens to grow proportional to N 2. These observations are true for both arrays of
Hertzian dipoles and isotropic radiators. The most noticeable
difference happens to be that the maximum array gain for the
case of Hertzian dipoles is a little bit below the one achieved
by isotropic radiators. For antenna separation larger than half
the wavelength, the array gain is much lower than with closely
spaced antennas, and displays an oscillatory nature. Again this
is true for both Hertzian dipoles and isotropic radiators. From
what we see in Figure 3, we can conclude that the transmit array gain of a ula of Hertzian dipoles is qualitatively the same
as if the array was equipped with isotropic radiators. This property clearly justifies the use of – mathematically and conceptually much simpler – isotropic radiators for theoretical treatment of antenna arrays.
A field-theoretic justification for the applicability of isotropic
radiators in antenna array modeling is given. A uniform linear
array of isotropic radiators leads qualitatively to the same relative antenna coupling as if Hertzian dipoles were used as the
array elements. Albeit isotropic antennas do not exist, their application in theory is legitimate, because they lead to a mutual
antenna coupling, which is reasonable from a physics point of
view. This allows for qualitatively correct analysis of important
performance parameters, like array gain.
6. REFERENCES
[1] J. C. Liberti and T. S. Rappaport, Smart Antennas for Wireless Communications. New Jersey: Prentice Hall, 1999.
[2] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge University Press, 2005.
[3] A. Balanis, Antenna Theory. Second Edition, John Wiley
& Sons, 1997.
[4] P. Russer, Electromagnetics, Microwave Circuit and Antenna Design for Communications Engineering, 2nd ed.
Nordwood, MA: Artec House, 2006.
[5] D. Kraus, Electromagnetics.
Hill, 1992.
Fourth Edition, McGraw-
[6] S. A. Schelkunoff and H. T. Friis, Antennas. Theory and
Practice. New York, NY: Wiley, 1952.
[7] M. T. Ivrlač and J. A. Nossek, “The Maximum Achievable
Array Gain under Physical Transmit Power Constraint,” in
Proc. IEEE International Symposium on Information Theory and its Applications, Dec. 2008, pp. 1338–1343.