Trade-off Between Optimal Efficiency and Envelope Correlation Coefficient for Antenna Clusters

Abstract

This paper introduces a theory for assessing and optimizing the multiple-input-multiple-output performance of multi-port cluster antennas in terms of efficiency, channel correlation, and power distribution. A method based on a convex optimization of feeding coefficients is extended with additional constraints allowing the user to control a ratio between the power radiated by the clusters. The formulation of the problem makes it possible to simultaneously optimize total efficiency and channel correlation with a fixed ratio between power radiated by the clusters, thus examining a trade-off between these parameters. It is shown that channel correlation, total efficiency, and allocation of radiated power are mutually conflicting parameters. The trade-offs are shown and discussed. The theory is demonstrated on a four-element antenna array and on a mobile terminal antenna.

Full text

1
Trade-off Between Optimal Efficiency and Envelope
Correlation Coefficient for Antenna Clusters
Vojtech Neuman, Miloslav Capek, Senior Member, IEEE, Lukas Jelinek, Anu Lehtovuori, and Ville Viikari,
Senior Member, IEEE
Abstract—This paper introduces a theory for assessing and
optimizing the multiple-input-multiple-output performance of
multi-port cluster antennas in terms of efficiency, channel cor-
relation, and power distribution. A method based on a convex
optimization of feeding coefficients is extended with additional
constraints allowing the user to control a ratio between the power
radiated by the clusters. The formulation of the problem makes it
possible to simultaneously optimize total efficiency and channel
correlation with a fixed ratio between power radiated by the
clusters, thus examining a trade-off between these parameters. It
is shown that channel correlation, total efficiency, and allocation
of radiated power are mutually conflicting parameters. The
trade-offs are shown and discussed. The theory is demonstrated
on a four-element antenna array and on a mobile terminal
antenna.
Index Terms—Antennas, electromagnetic theory, feeding opti-
mization, multiple-input-multiple-output, mutual coupling, con-
vex optimization.
I. INTRODUCTION
MULTIPLE-INPUT-MULTIPLE-OUTPUT (MIMO)
technology has proven its utility in wireless
communication in increasing data throughput and
coverage while mitigating signal fading due to multipath
propagation [1]. All contemporary wireless standards utilize
MIMO [2], and this trend is expected to continue [3], [4]. Of
high priority is the implementation of MIMO technology for
mobile devices at sub-6 GHz frequency bands [2]. Though
narrow bands at this frequency range cannot offer data rates
as high as millimeter-wave frequencies, they still have an
advantage in robustness against environmental influences and
free space loss [5], [6]. The small electric size of user devices
in these frequency bands nevertheless significantly degrades
their efficiency and channel correlation performance, two key
parameters affecting MIMO system performance [1], [7].
When speaking of efficiency we refer to an antenna’s ability
to transform incoming guided power into radiated power [8].
In the current state of the art, approaches exist for efficiency
enhancement [9], [10]. Nevertheless, they solely deal with the
Manuscript received May 3, 2023; revised May 3, 2023. This
work was supported by the Czech Science Foundation under
project No. 21-19025M and by the Czech Technical University in Prague
under project SGS22/162/OHK3/3T/13.
V. Neuman, M. Capek and L. Jelinek are with the Czech Technical
University in Prague, Prague, Czech Republic (e-mails: {vojtech.neuman;
miloslav.capek; lukas.jelinek}@fel.cvut.cz).
A. Lehtovuori and V. Viikari are with the Aalto University, Espoo, Finland
(e-mails: {anu.lehtovuori; ville.viikari}@aalto.fi).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org
efficiencies of individual channels, i.e., one antenna element
is fed, and the rest of the elements are terminated into the
given impedance [11]. The efficiency is evaluated, measured,
or optimized only under this feeding scenario. Although this
method is simple, when the system is not well-isolated, an
input impedance of a given port heavily depends on the
feeding of other ports of a multi-port system. The efficiencies
derived in this way may not reflect the realized efficiencies
under active (simultaneous) feeding of all ports. The method
introduced in this paper enables the versatile design of a
multi-antenna system and avoids above mentioned issues by
considering only the total efficiency of the whole radiating
system which means that the correct input impedance of all
ports concerning chosen feeding is taken into account.
Channel correlation is closely associated with mutual cou-
pling between antenna ports [7], [12]. Several approaches
were introduced to reduce it [9], [13], e.g., defective ground
surfaces [14]–[16], electromagnetic band-gap structures [13],
decoupling networks [17], utilization of characteristic modes
theory [18]–[20], or employment of point symmetries [21],
[22]. These techniques have proven their ability to reduce
mutual coupling. However, they cannot optimize the system’s
total efficiency simultaneously.
The method applied in this paper is based on grouping
more antenna elements into so-called antenna clusters [23].
The mutual coupling between elements within one cluster is
employed to increase its performance. The antenna cluster
consists of multiple elements, each having its ports and proper
feeding coefficients at every frequency to ensure optimal
performance. A distributed transmitter supports this operation
principle [24]. Replicating antenna clusters within the design
region leads to MIMO functionality [25]. Although mutual
coupling is beneficial between elements of one antenna cluster,
it is still a potential issue in the MIMO system [25], i.e.,
between different clusters.
This paper aims to introduce the simultaneous optimization
of total efficiency and channel correlation, constrained by
a user-defined allocation of power radiated by the clusters.
This is achieved by employing antenna cluster feeding op-
timization [25], [26] supplemented by additional constraints
for controlling channels radiating power and power interfering
between channels. The optimization is solely based on the
utilization of a matrix description. This work additionally
proposes a theory of power ratios where the radiated power
is divided into individual components describing the power
composition. The above allows the analysis of a trade-off
between efficiency, radiated power by channels, and channel
arXiv:2305.01416v1 [cs.IT] 2 May 2023

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P12
P21
P11
P22
(a) (b)
individual cluster system of two clusters
Fig. 1. Illustration of an antenna cluster. The indicated integrated circuit
represents a distributed transceiver. (a) Multi-port antenna forming an antenna
cluster (two feeding ports, two radiating elements). (b) Two antenna clusters
forming an antenna system. The wavy curves and arrows depict self and
mutual radiated powers.
correlation.
The organization of this paper is as follows. Section II
introduces the mathematical tools used. The modified opti-
mization procedure is described in Sec. III, and its application
is shown in Sec. IV and Sec. V. The paper concludes in Sec VI.
Appendices A and B reviews the theoretical background, and
Appendices C and D deal with further aspects of the developed
theory.
II. RA DI ATIO N OF AN TE NNA CLUS TE RS
An example of an antenna cluster is shown in Fig. 1(a). Mul-
tiple antenna clusters establish a radiating system (MIMO),
see Fig. 1(b), where two antenna clusters form two individual
communication channels.
The Nports of the radiating body are divided into M
antenna clusters, where the m-th antenna cluster has Nm
assigned ports
N=
M
X
m=1
Nm.(1)
The number Malso defines the order of the MIMO used [1].
The radiating system is fully described with its admittance
matrix y∈CN×N, see Appendix A, which, together with port
voltages, calculate cycle mean power
1
2vHyv =1
2vHg0v+i
2vHb0v=P+ iPreact,(2)
where g0is the port radiation matrix, b0is the port sus-
ceptance matrix, Pdenotes radiated power, Preact represents
reactive power and v∈CN×1aggregates feeding voltages
connected to the antenna ports. Considering the far-field
correlation, see Appendix B, the port radiation matrix g0is the
operator of primary interest1. Individual elements of radiation
matrix g0are grouped into blocks as
g0=


g0,11 . . . g0,1M
.
.
.....
.
.
g0,M1. . . g0,MM


,(3)
with g0,mm ∈CNm×Nmdescribing the interaction within the
m-th cluster and g0,mn ∈CNm×Nnconsidering interactions
between the m-th and n-th cluster. The radiation matrix is
hermitian
g0,nm =gH
0,mn.(4)
For a simplification of the matrix description, the indexing
operators Cm∈BN×Nmare introduced as
Cm,kl =(1the k-th port is the m-th cluster l-th port,
0otherwise.
(5)
Applying this operator from both sides on the port-mode
radiation matrix gives
g0,mn =CH
mg0Cn.(6)
Using the combination of CmCH
mon the same group of
operators provides a matrix that reads
e
g0,mn =CmCH
mg0CnCH
n=


0. . . 0
.
.
.g0,mn
.
.
.
0. . . 0


.(7)
Those two principles are frequently used in the following.
The feeding of an antenna is also partitioned into blocks,
each representing a feeding vector of a particular cluster
v=vT
1. . . vT
MT,(8)
where v∈CN×1contains vectors vm∈CNm×1correspond-
ing to the m-th cluster feeding.
The radiated power consists of terms
Pmn =1
2vHCmCH
mg0CnCH
nv=1
2vH
mg0,mnvn,(9)
where m=nrepresents self terms, and m6=nrepresents
interaction terms. Summing all contributions gives the total
radiated power
P=1
2X
m
vH
mg0,mmvm+1
2X
mX
n6=m
vH
mg0,mnvn.(10)
Notice that only total radiated power Pin (10) has a clear
physical meaning. The separation into generally complex-
valued terms (9) is only used to exploit their mathematical
properties for the optimization procedure discussed in Sec. III.
It is nevertheless worth mentioning that power terms Pmn are
directly related to the far-field correlation coefficient, see (35)
in Appendix B.
1Although the lossless antennas are used for simplicity, lossy cases can be
included without any major modification of the theory, see Appendix A.

3
Dividing relation (10) by total radiated power Pleads to
power ratios αmn
1 = X
m
αmm +X
mX
n6=m
αmn =
X
m
αmm +X
mX
n>m
βmn,(11)
where αmn =Pmn/P and βmn =αmn +αnm . Power ratios
αmn with m6=nhave an imaginary part, for which γmn is
defined as
γmn = i (αmn −αnm).(12)
The power ratios are used in the optimization to control
the composition of the radiated power. The self-term power
ratios αmm control useful radiated power, and βmn, γmn are
used to limit the mutual radiated power. More details are given
in Appendices C and D.
III. OPTIMIZATION PR OBLEM AND CONSTRAINTS
The trade-off between far-field correlation, total efficiency,
and power allocation is formulated in this section, starting with
simpler sub-problems.
The optimization problem for maximum total efficiency η
reads
maximize
v
P
subject to Pin = 1,(13)
and it can be solved as a generalized eigenvalue problem [26].
This formulation delimits the highest total efficiency achieved
with a given set of ports. To constrain optimization, the
power delivered to the antennas is fixed. Depending on the
knowledge of the feeding system, the delivered power can be
determined as available power [27], [28] or input power [29],
see Appendix A. The problem (13) is unable to distinguish
individual channels. Therefore, a modification in the form of
additional constraints is required.
Observing the formula for the far-field correlation (35), it is
clear that the term in its numerator should be minimized. The
terms of form Pmn with m6=nare generally complex and
unsuitable for convex solvers. Thus real mutual radiated power
Pmn,real is introduced and defines a constraint on mutual
radiated power
Pmn,real =Pmn +Pnm =βmnP. (14)
where βmn is the power ratio from (11) the value of which is
set by the user during optimization.
Constraint (14) deals only with the real part of the cor-
relation coefficient (35). The imaginary part also has to be
suppressed. A constraint on imaginary mutual power Pmn,imag
reads
Pmn,imag = i (Pmn −Pnm) = γmn P(15)
where γmn is used in a similar manner as βmn.
The trivial solution for zeroing channel correlation is to set
either vmor vnto zero vector. Nevertheless, this would lead to
a situation in which Pmm or Pnn equals zero. Such a scenario
is unacceptable since it disables one of the clusters. Hence,
another set of constraints must be defined. To that point, self-
radiated powers Pmm are constrained via
Pmm =αmmP, (16)
stating that the channel radiates a user-defined ratio αmm of
total radiated power.
Considering the constraints defined above, the modified
optimization problem reads
maximize
v
P
subject to Pin = 1,
Pmm =αmmP, ∀m
Pmn,real =βmnP, ∀m, n ∧m6=n,
Pmn,imag =γmnP, ∀m, n ∧m6=n,
(17)
This definition fully controls radiated power optimization and
far-field correlation while maximizing total efficiency.
Before showing the described theory in an example, it is
worth pointing out the following remarks. The system with M
clusters needs, in total, M2constraints2. This number becomes
considerably high even for a small number of antenna clusters.
In practice, however, some constraints might be omitted. For
example, the channel correlation between particular clusters
can be already sufficiently low due to a geometrical arrange-
ment.
Although the first choice of the power ratios in problem (17)
might be αmm = 1/M and βmn, γmn →0, it is beneficial
to perform a sweep over these values. The sweep not only
provides an understanding of how radiated power is dis-
tributed among the channels, but it might also offer important
relaxations that would be compensated by gains in overall
performance.
Throughout this paper, the solution of problem (17) is
approached using a dual formulation [30] and QCQP solvers
contained within the fundamental bounds package [31]. When
a duality gap [30] is present, the interior-point method [32]
applied directly to the primal problem is utilized instead. In
these cases, the localization of a global optimum is not assured.
A comparison with the Monte Carlo method suggests that
these extremal points might be globally optimal.
IV. EXA MP LE : PARALLEL DIPOLES
To demonstrate the most salient features of the proposed
optimization, four thin parallel dipoles made of a perfect elec-
tric conductor are used, see Fig. 2. The length of each strip is
equal to L= 0.916 c0/(2f0), where f0= 750 MHz. Spacing
between dipoles is equal to L/4. All ports are connected to the
50 Ω transmission line, and no additional components are used
for matching. Figure 2 also shows how antenna elements are
grouped into two clusters. The in-house method of moments
solver [33] is employed for the system analysis.
2The given number respects the dependence of power components.

4
cluster 1
cluster 2
L
L/4
L/100
x
z
yxz
×
Fig. 2. Illustration of the considered MIMO system. The dimensions are not
shown in the correct proportions. The blue and red curves highlight the first
and second clusters. All dipoles are fed with delta gaps in places highlighted
with orange lines.
The optimization problem reads
maximize
v
P
subject to Pin = 1,
P11 =α11P,
P12,real =β12P,
P12,imag =γ12P,
(18)
where α11, β12 , and γ12 are swept. In practical scenarios, a
given ratio κ12 between P11 and P22 is typically of interest.
Utilizing the dependence (11) and defining a power ratio κ12
as
κ12 =P22
P11
=α22
α11
(19)
allows fixing the power ratio κ12 by using
α11 =1−β12
1 + κ12
,(20)
thereby reducing the sweep to only two dimensions. The rest
of this section is dedicated to solving problem (18) on different
frequencies with various power ratio settings to show different
features of the developed theory.
The results of the two optimization problems are compared:
the maximization of total efficiency ηin problem (13) and the
simultaneous minimization of the envelope correlation coeffi-
cient E12 in problem (18). In the latter case, the optimization
is solved using κ12 = 1,β12 = 0 and γ12 = 0,i.e., enforcing
the zero correlation coefficient and enforcing equal power
distribution among the channels. Considering the properties
of constrained optimization [30], [32], the expected result is a
lower value of total efficiency for constrained problem (18)
as compared to the former problem (13). Figure 3 shows
that optimization of total efficiency leads to a relatively large
envelope correlation coefficient (E12 >0.5) in most of the
studied frequencies. When the constraint on E12 is added, total
efficiency is slightly decreased, but the envelope correlation
700 720 740 760 780 800
0
0.2
0.4
0.6
0.8
1
Eq. η E12
(13)
(18)
f[MHz]
η[−], E12 [−]
Fig. 3. The frequency sweep of total efficiency ηand envelope correlation
coefficient E12 for κ12 = 1,β12 = 0 and γ12 = 0. The blue color
represents the former optimization problem (13), and the red is its modified
version (18). The jump in the blue dashed curve is caused by the change of
feeding eigenvector vto a different mode.
−0.5−0.25 0 0.25 0.5
−0.4
−0.2
0
0.2
0.4
0.6
0.8
P
−0.42
−0.082
0.46
0.84
γ12 [−]
Px[W]
P11 Preal,12
P22 Pimag,12
Fig. 4. The sweep of power ratio γ12 at f= 721 MHz with β12 =−0.1,
α11 = 0.55 and α22 = 0.55. The input power reads Pin = 1 W. The
black curve denotes total radiated power P. The rest of the curves depict its
decomposition into individual parts (10).
coefficient is kept at zero guaranteeing acceptable MIMO
performance.
Various power ratio settings are further studied in Fig. 4
which demonstrates the situation in which problem (18) is
solved at f= 721 MHz with κ12 = 1,β12 =−0.1and
γ12 ∈[−0.5,0.5]. It shows how different power compo-
nents (10) are related to total radiated power Pthrough power
ratios. The curves for P11 and P22 overlay each other as the
power ratio reads κ12 = 1.
Evaluating the trade-off between total efficiency ηand enve-
lope correlation coefficient E12, the power ratio sweep is ex-
tended to β12, γ12 ∈[−0.5,0.5], see Fig. 5, and problem (18)
is solved at f= 735 MHz with κ12 = 10. Total efficiency is
depicted as a contour plot and envelope correlation is shown
with ellipses of constant E12, see Appendix C. The figure
also shows that not all combinations of power ratios lead to a
feasible solution. The unfeasible solutions are bounded by a
maximal envelope correlation ellipse. Optimal total efficiency
and the optimal envelope correlation coefficient lie at different
spots in the β12 ×γ12 space, showing that they are conflicting

5
0.4
0.2
0.5
0.3
0.1
−0.5−0.25 0 0.25
−0.5
−0.25
0
0.25
0.5
β12 [−]
γ12 [−]
0.55
0.62
0.69
0.76
0.83
0.91
η[−]
Fig. 5. The power ratio sweep β12 ∈[−0.5,0.5] and γ12 ∈[−0.5,0.5]
for power ratio κ12 = 10. The solid-line contour plot depicts total efficiency.
Dashed ellipses are regions of constant envelope correlation E12. The orange
dots refer to Pareto-optimal solutions Fig. 6.
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
1
E12 [−]
η[−]
κ= 1
κ= 10
Fig. 6. The trade-off between the envelope correlation coefficient and total
efficiency for two different ratios κ12 between self powers. Pareto optimal
curves are highlighted. Dots represent randomly generated solutions.
parameters.
The trade-off between total efficiency ηand envelope cor-
relation coefficient E12 at frequency f= 735 MHz is shown
in Fig. 6. The blue and red continuous curves are Pareto
optimal solutions. A feeding vector with good efficiency is not
a guarantee of an acceptable envelope correlation coefficient.
The blue and red dots represent randomly generated solutions
fulfilling constraints on input power and power ratio κ12.
Random solutions lie under the Pareto frontiers, proving the
validity of the presented method. In this specific case, there
is only a slight difference in performance between different
power ratios κ12. Also, the drop in efficiency for lower
envelope correlation coefficients is minor.
The radiation patterns for κ12 = 1,β=−0.045 and γ= 0,
see Fig. 3, at frequency 750 MHz are depicted in Fig. 7.
The pattern of the whole system corresponding to voltage
vector v=vT
1vT
2Tsolving problem (18) is depicted
in purple. This radiation pattern is composed of individual
patterns of each cluster (blue and red), which correspond
to voltage vectors v=vT
10TTand v=0TvT
2T,
respectively, and resemble cardioids with the opposite orien-
0◦
−30◦
−60◦
−90◦
−120◦
−150◦
180◦
150◦
120◦
90◦
60◦
30◦
0.9
1.8
2.7
3.6
4.5
0◦
−30◦
−60◦
−90◦
−120◦
−150◦
180◦
150◦
120◦
90◦
60◦
30◦
0.9
1.8
2.7
3.6
4.5
(a) θ-plane (b) φ-plane
Fig. 7. Total and partial radiation patterns resulting from the solution to (18)
for κ12 = 1,β=−0.045 and γ= 0 at central frequency f0. The red and
blue colors represent each channel. (a) cut in ϕ= 0. (b) cut in θ=π/2.
75
62
44
52.8
48.8
150
20
10.8
9
x
y
z
cluster 1cluster 2
Fig. 8. Simplified model of a mobile terminal. The blue and red curves
highlight the first and second clusters. Monopole antennas are fed with
delta gaps in places highlighted with orange lines. The dimensions are in
millimeters.
tation of their main lobes, a sound solution for high efficiency
and low far-field correlation.
V. EXAMPLE: MOBILE TER MI NAL
The second studied structure resembles a mobile termi-
nal introducing a more realistic antenna example with size
restricted to common present-day devices, see Fig. 8. The
antenna system is considered for operations in the vicinity
of 900 MHz. All ports are connected to 50 Ω transmission
lines and no additional components are used for matching.
The presented system is purposely not symmetric. The four
antenna elements are grouped into two clusters as indicated in
Fig. 8.
Enforcing the vanishing envelope correlation coefficient is,
generally, not the best strategy for optimization due to its
price in total efficiency. Instead, it is beneficial to add an

6
800 850 900 950 1000
0
0.2
0.4
0.6
0.8
1
Emax η E12
0.25
0
f[MHz]
η[−], E12 [−]
Fig. 9. The frequency sweep of total efficiency ηand the envelope correlation
coefficient E12 for κ12 = 1. The red color represents the optimization
problem for E12 = 0, and the blue curve is the problem with envelope
correlation coefficient E12 ≤0.25.
additional non-equality constraint that sets a tolerance for
envelope correlation coefficient E12 under a given threshold
maximize
v
P
subject to Pin = 1,
P11 =α11P,
P12,real =β12P,
P12,imag =γ12P,
E12 ≤Emax,
(21)
where Emax is the maximal allowed envelope correlation
coefficient.
Figure 9 shows the solution to problem (21) where the
set of red curves corresponds to Emax = 0 and the blue
curves represent a solution for Emax = 0.25. The cost of
zero envelope correlation coefficient E12 is a significant drop
in total efficiency ηin the entire frequency region. Allowing
a higher envelope correlation coefficient than zero (always
the case in practical applications) leads to substantially better
results in total efficiency.
The trade-off between total efficiency ηand envelope cor-
relation coefficient E12 is shown in Fig. 10 for f= 960 MHz
and for two different power ratios κ12. The maximum allowed
value of correlation Emax is not limited in this case. The
continuous lines are Pareto frontiers, while the dots represent
randomly generated solutions. This example highlights the
conflict between the power ratio and total efficiency caused
by the lack of symmetries. Unlike the previous example,
this antenna model has a significant trade-off between total
efficiency and the envelope correlation coefficient.
The distributed transceiver implementation [24] limits the
deployment of antenna cluster technology. Optimal feeding
cannot be applied to all frequencies simultaneously due to
physical realization restrictions [24], it would have to be
discretized into narrower sub-bands. This introduces a trade-
off of how many frequency points should be used to conserve a
desired performance. Addressing this problem leads to treating
the bandwidth of optimal feeding coefficients near a single
frequency. Figure 11 shows the difference in performance
0 0.1 0.2 0.3 0.4 0.5 0.6
0.1
0.3
0.5
0.7
0.9
E12 [−]
η[−]
κ= 1
κ= 10
Fig. 10. Trade-off between the envelope correlation coefficient and total
efficiency for the mobile terminal model at frequency f= 960 MHz. Two
different ratios κ12 between powers are shown. The blue and red dots are
randomly generated solutions from the Monte Carlo method. Pareto-optimal
curves are highlighted by the solid lines.
845 855 865 875 885
0
0.2
0.4
0.6
0.8
1
1.2
v∗
v(fc)
η
E12
κ12
f[MHz]
η[−], E12 [−], κ12 [−]
Fig. 11. Comparison of solutions to problem (21) with Emax = 0.25. Solid
lines show the performance of voltage vector voptimized at each frequency.
Dashed lines correspond to the voltage vector that is only optimal at frequency
fc= 867 MHz (highlighted by the black dashed vertical line).
for the case when the optimal solution to problem (21) is
applied to all frequencies in a given sub-band and for the case
where the optimal solution for fc= 867 MHz is applied to all
considered frequencies. Total efficiency η, envelope correlation
coefficient E12, and power ratio κ12 =P11/P22 at fcare equal
for both excitation schemes. All metrics gradually change as
the distance from fcincreases. To summarize, the optimal
voltages do not deviate significantly when the frequency is
changed, though the effect of frequency detuning must be
evaluated in more detail in the future, also keeping in mind
that feeding transmission lines between the feeding circuitry
and the antenna port would result in additional dispersion.
VI. CONCLUSION
The theory of antenna cluster optimization has been ex-
panded toward evaluating the trade-off between total effi-
ciency, envelope correlation coefficients, and power radiated
by each cluster. The trade-off is evaluated via a solution to
the quadratic optimization problem for maximization of total
efficiency extended with the capability to deal simultaneously

7
with the three mentioned parameters. This is enabled by
introducing ratios between the power radiated by the clusters
and radiated power component separation. The series of results
in the two examples validate the proposed theory with the
second example highlighting the trade-off between the three
mentioned parameters. The most significant outcome of this
paper is to demonstrate the ability of an antenna cluster to
balance the level of far-field pattern correlation and total
efficiency.
The optimization procedure yields optimal voltages which
provide Pareto-optimal performance when connected to the
antenna’s ports. Applying optimal voltages to all frequencies
leads to an unattainable complexity of feeding circuitry. There-
fore, the distributed transceiver needs to operate in sub-bands
where one solution is applied. The feeding circuitry is also
limited to discrete levels of feeding coefficients magnitudes
and phases, which can further affect the performance, and with
dispersion caused by transmission lines between the distributed
transceiver and antenna ports. This work did not assume
any form of matching circuits for further total efficiency
enhancement. This is an additional set of degrees of freedom
to be considered by an antenna designer.
An unresolved issue is the occasional appearance of a
duality gap when the dual formulation is employed to solve the
optimization problem. The employment of different solving
schemes should also be investigated to guarantee a glob-
ally optimal solution. This problem will likely be enhanced
when more than two antenna clusters are studied. Lastly,
the presented investigation raised the question concerning the
minimal number of ports required to guarantee the desired
performance in all optimized metrics.
APPENDIX A
MATRIX OPE RATORS AND PORT QUANTITIES
The considered radiating system is described by the
impedance matrix Z=R0+ iX0extracted from any integral
equations [34], [35] method
ZI =V,(22)
where Iaggregates current approximation coefficients and V
is a discrete form of an excitation [36]. The antenna metrics are
commonly expressed as linear and quadratic forms of current
vector I. Notable examples used in this paper are radiated P
and reactive Preact powers
1
2IHZI ≈P+ iPreact.(23)
This paper deals with only lossless structures. However,
the losses can be involved by adding loss matrix Rρinto
impedance matrix Zwithout any other changes.
Two operators are defined for the purposes of transforming
into port modes [26] with the first reading
Dmn =(ζmm=n,
0otherwise,(24)
where ζmis a parameter used to control units of port quan-
tities. The second operator is a port indexing matrix defined
as
Nmn =(1the n-th port is placed at the m-th position,
0otherwise.
(25)
Current vector Ican now be controlled with voltage sources
at antenna ports as
I=YDNv,(26)
where admittance matrix Yis an inverse of the Zoperator.
Relation (26) can be used to transform MoM matrices Mto
their port equivalents
m=NHDHYHMYDN.(27)
As an example, relation (27) can be used to transform radiation
operator R0from (23) to port radiation matrix
g0=NHDHYHR0YDN,(28)
with the help of which the radiated power can be evaluated as
P≈1
2vHg0v.(29)
Another essential quantity is the power delivered into the
radiating system which is used to constrain the optimization
problem. Available power [27] can be determined based on the
knowledge of the admittance matrix [28] of the feeding sys-
tem. If the signal generators are uncoupled and connected to
the antennas with transmission lines of real-valued impedance,
available power is equal to incident power [29], which can be
determined as
Pin =1
2aHa.(30)
To make it suitable for the optimization considered in this
paper, incoming power waves aare related to port voltages v
as [26]
a=kiv=1
2Λ−1+ Λy0v.(31)
Finally, a combination of (29) and (30) leads to total
efficiency [26]
η=IHR0I
aHa=vHg0v
vHkH
ikiv,(32)
which is the metric to be maximized in this text.
APPENDIX B
FAR-FIELD CORRELATION
Mutual radiated power Pmn, typically used to quantify far-
field orthogonality, can be evaluated as
Pmn =1
2Z0I
S
F∗
m(θ, ϕ)·Fn(θ, ϕ) dΩ, (33)
where indices m, n denote current vectors Im,Inexcited on
an antenna. Utilizing method of moments together with (29),
this can be written as a quadratic form
Pmn =1
2IH
mR0In=1
2vH
mg0,mnvn,(34)

8
with the help of which a far-field correlation coefficient ρmn
is expressed as
ρmn =vH
mg0,mnvn
pvH
mg0,mmvmpvH
ng0,nnvn
.(35)
Although the generally complex coefficient (35) is not
directly suitable for convex optimization, individual quadratic
forms in the nominator and denominator can be added to the
optimization problem as additional constraints.
The envelope correlation coefficient Emn reads
Emn ≈ |ρmn|2.(36)
APPENDIX C
COR RE LATI ON EXPRESSED WITH POW ER RATIOS
This section expresses correlation through power ratios.
Considering a case of two antenna clusters M= 2, the far-
field correlation coefficient reads
ρ12 =vH
1g0,12v2
pvH
1g0,11v1pvH
2g0,22v2
=1
2
β12 + iγ12
√α11√α22
,(37)
which is obtained by dividing every term vH
mg0,mnvnwith
vHg0vand utilizing the power ratios. The ratio between self
powers κ12 =P11/P22 is usually of interest. Using (11) and
ratio κ12 allows us to express α11 as
α11 =1−β12
1 + κ12
,(38)
and α22 =κ12α11 . Putting (38) into relation (37) leads to
E12 =1 + κ12
2√κ12 2β2
12 +γ2
12
(β12 −1)2.(39)
The first term in parentheses in (39) is the scaling factor and
the rest of the expression forms the equation for the ellipse.
Considering κ12 = 1 for simplicity, then the expression (39)
can be rewritten as
(1 −E12)2
E12 β12 +E12
1−E12 2
+1−E12
E12
γ2
12 = 1,(40)
which is an ellipse equation. The general κ12 would need
only a re-scaling of E12 by the factor in parentheses in
equation (39).
APPENDIX D
POWE R RATIOS INTERVA LS
The radiated power ratios cannot be arbitrary. The feasible
intervals for these quantities can be obtained using generalized
eigenvalue problems. Starting with self powers, the maximal
value of αmm is found as
e
g0,mmvj=αjg0vj,(41)
and the corresponding interval reads α∈[0,max αj].
A similar approach is used for mutual power ratios. The
generalized eigenvalue problems read
(e
g0,mn +e
g0,nm)vk=βkg0vk,(42)
i (e
g0,mn −e
g0,nm)vl=γlg0vl,(43)
and determine intervals βmn ∈[min βk,max βk]for the real
part and γmn ∈[min γl,max γl]for the imaginary part of
mutual powers.
−2−1.5−1−0.5 0
−1
−0.5
0
0.5
1
β12 [−]
γ12 [−]
0
0.1
0.2
0.3
0.4
0.5
E12 [−]
Fig. 12. Depiction of envelope correlation expressed with power ratios for
the case of two antenna clusters and κ12 = 1. Dashed ellipses show curves of
the constant envelope correlation coefficient. Values higher than E12 >0.5
are not shown in compliance with [2].
REFERENCES
[1] A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-time Wireless
Communications. Cambridge University Press, 2003.
[2] (2023, Apr.) 3rd Generation Partnership Project (3GPP) . 3GPP.
[Online]. Available: https://www.3gpp.org/
[3] M. Giordani, M. Polese, M. Mezzavilla, S. Rangan, and M. Zorzi,
“Toward 6G Networks: Use Cases and Technologies,” IEEE Commu-
nications Magazine, vol. 58, no. 3, pp. 55–61, 2020.
[4] I. F. Akyildiz, A. Kak, and S. Nie, “6G and Beyond: The Future of
Wireless Communications Systems,” IEEE Access, vol. 8, pp. 133995–
134 030, 2020.
[5] A. Ghasemi, A. Abedi, and F. Ghasemi, Propagation Engineering in
Wireless Communications, 2nd ed. Springer Cham, Jun. 2018.
[6] M. Shafi, J. Zhang, H. Tataria, A. F. Molisch, S. Sun, T. S. Rappaport,
F. Tufvesson, S. Wu, and K. Kitao, “Microwave vs. Millimeter-Wave
Propagation Channels: Key Differences and Impact on 5G Cellular
Systems,” IEEE Communications Magazine, vol. 56, no. 12, pp. 14–
20, 2018.
[7] R. G. Vaughan and J. B. Andersen, “Antenna Diversity in Mobile
Communications,” IEEE Transactions on Vehicular Technology, vol. 36,
no. 4, pp. 149–172, 1987.
[8] C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. Wiley,
1996.
[9] S. Kumar, A. S. Dixit, R. R. Malekar, H. D. Raut, and L. K. She-
vada, “Fifth Generation Antennas: A Comprehensive Review of Design
and Performance Enhancement Techniques,” IEEE Access, vol. 8, pp.
163 568–163 593, 2020.
[10] J. Park, M. Jeong, N. Hussain, S. Rhee, S. Park, and N. Kim,
“A Low-Profile High-Gain Filtering Antenna for Fifth Generation
Systems Based on Nonuniform Metasurface,” Microwave and Optical
Technology Letters, vol. 61, no. 11, pp. 2513–2519, 2019. [Online].
Available: https://onlinelibrary.wiley.com/doi/abs/10.1002/mop.31931
[11] S. Stein, “On Cross Coupling in Multiple-Beam Antennas,” IRE Trans-
actions on Antennas and Propagation, vol. 10, no. 5, pp. 548–557, 1962.
[12] S. Blanch, J. Romeu, and I. Corbella, “Exact Presentation of Antenna
System Diversity Performance From Input Parameter Description,” Elec-
tronics Letters, vol. 39, pp. 705–707, 06 2003.
[13] I. Nadeem and D.-Y. Choi, “Study on Mutual Coupling Reduction
Technique for MIMO Antennas,” IEEE Access, vol. 7, pp. 563–586,
2019.
[14] M. Ayatollahi, Q. Rao, and D. Wang, “A Compact, High Isolation and
Wide Bandwidth Antenna Array for Long Term Evolution Wireless
Devices,” IEEE Transactions on Antennas and Propagation, vol. 60,
no. 10, pp. 4960–4963, 2012.
[15] R. A. Bhatti, J.-H. Choi, and S.-O. Park, “Quad-Band MIMO Antenna
Array for Portable Wireless Communications Terminals,” IEEE Anten-
nas and Wireless Propagation Letters, vol. 8, pp. 129–132, 2009.
[16] C.-Y. Chiu, C.-H. Cheng, R. D. Murch, and C. R. Rowell, “Reduction
of Mutual Coupling Between Closely-Packed Antenna Elements,” IEEE
Transactions on Antennas and Propagation, vol. 55, no. 6, pp. 1732–
1738, 2007.

9
[17] S.-C. Chen, Y.-S. Wang, and S.-J. Chung, “A Decoupling Technique for
Increasing the Port Isolation Between Two Strongly Coupled Antennas,”
IEEE Transactions on Antennas and Propagation, vol. 56, no. 12, pp.
3650–3658, 2008.
[18] R. Martens and D. Manteuffel, “Systematic Design Method of a Mobile
Multiple Antenna System Using the Theory of Characteristic Modes,”
IET Microwaves, Antennas Propagation, vol. 8, no. 12, pp. 887–893,
2014.
[19] B. Yang and J. J. Adams, “Systematic Shape Optimization of Symmetric
MIMO Antennas Using Characteristic Modes,” IEEE Transactions on
Antennas and Propagation, vol. 64, no. 7, pp. 2668–2678, 2016.
[20] R. Martens and D. Manteuffel, “2-port Antenna Based on the Selective
Excitation of Characteristic Modes,” in Proceedings of the 2012 IEEE
International Symposium on Antennas and Propagation, 2012, pp. 1–2.
[21] N. Peitzmeier and D. Manteuffel, “Upper Bounds and Design Guidelines
for Realizing Uncorrelated Ports on Multimode Antennas Based on
Symmetry Analysis of Characteristic Modes,” IEEE Transactions on
Antennas and Propagation, vol. 67, no. 6, pp. 3902–3914, 2019.
[22] M. Masek, L. Jelinek, and M. Capek, “Excitation of Orthogonal Radia-
tion States,” IEEE Transactions on Antennas and Propagation, vol. 69,
no. 9, pp. 5365–5376, 2021.
[23] J. M. Hannula, T. Saarinen, J. Holopainen, and V. Viikari, “Frequency
Reconfigurable Multiband Handset Antenna Based on a Multichannel
Transceiver,” IEEE Transactions on Antennas and Propagation, vol. 65,
no. 9, pp. 4452–4460, 2017.
[24] A. R. Saleem, K. Stadius, J.-M. Hannula, A. Lehtovuori, M. Kosunen,
V. Viikari, and J. Ryyn¨
anen, “A 1.5–5-GHz Integrated RF Transmitter
Front End for Active Matching of an Antenna Cluster,” IEEE Transac-
tions on Microwave Theory and Techniques, vol. 68, no. 11, pp. 4728–
4739, 2020.
[25] J.-M. Hannula, T. Saarinen, A. Lehtovuori, J. Holopainen, and V. Viikari,
“Tunable Eight-Element MIMO Antenna Based on the Antenna Cluster
Concept,” IET Microwaves, Antennas and Propagation, vol. 13, Feb.
2019.
[26] M. Capek, L. Jelinek, and M. Masek, “Finding Optimal Total Active
Reflection Coefficient and Realized Gain for Multiport Lossy Antennas,”
IEEE Transactions on Antennas and Propagation, vol. 69, no. 5, pp.
2481–2493, 2021.
[27] C. Desoer, “The Maximum Power Transfer Theorem for N-ports,” IEEE
Transactions on Circuit Theory, vol. 20, no. 3, pp. 328–330, 1973.
[28] F. Broyd´
e and E. Clavelier, “The Radiation and Transducer Efficiencies
of a Multiport Antenna Array,” Excem Res. Papers in Electron.
and Electromagn., no. 4, pp. 1–22, Jan. 2022. [Online]. Available:
https://doi.org/10.5281/zenodo.5816837
[29] D. M. Pozar, Microwave Engineering, 3rd ed. Wiley, 2005.
[30] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge
University Press, 2004.
[31] J. Liska, L. Jelinek, and M. Capek, “Fundamental Bounds to Time-
Harmonic Quadratic Metrics in Electromagnetism: Overview and Im-
plementation,” 2021.
[32] J. Nocedal and S. Wright, Numerical Optimization, 2nd ed., ser. Springer
series in operations research. Springer, 2006.
[33] (2023, Apr.) Antenna Toolbox for MATLAB (AToM). Czech Technical
University in Prague. [Online]. Available: www.antennatoolbox.com
[34] J. L. Volakis and K. Sertel, Integral Equation Methods for Electromag-
netics. SciTech Publishing, 2012.
[35] W. C. Chew, M. S. Tong, and B. Hu, Integral Equation Methods for
Electromagnetic and Elastic Waves, 1st ed., ser. Synthesis Lectures on
Computational Electromagnetics. Morgan and Claypool Publishers,
2007.
[36] R. F. Harrington, Field Computation by Moment Methods, ser. The IEEE
PRESS Series in Electromagnetic Waves (Donald G. Dudley, Editor).
Wiley-IEEE Press, 1993.