IEICE TRANS. COMMUN., VOL.E99–B, NO.1 JANUARY 2016

95

PAPER

Special Section on Recent Progress in Antennas, Propagation and Wireless Systems Related to Topics in ISAP2014

Channel Capacity Evaluation of MIMO Antenna Based on
Eigenvalues of S-Parameter
Naoki HONMA†a) , Kentaro MURATA†† , Hiroshi SATO††† , Members, Koichi OGAWA†††† , Senior Member,and Yoshitaka TSUNEKAWA† , Member

SUMMARY In this paper, a method of calculating the mean channel
capacity based on S -parameter of MIMO (Multiple-Input Multiple-Output)
antenna is proposed. This method exploits the correlation matrix calculated
from the antenna S -parameter matrix, and oﬀers highly accurate estimatesof the mean channel capacity without dependence on SNR (Signal-to-Noise
Ratio). The numerical and experimental results revealed that the proposed
method can calculate the channel capacity with fair accuracy independentof the number and spacing of the antenna elements if the radiation eﬃciency is suﬃciently high.
key words: MIMO, capacity, S -parameter

1. Introduction
Evaluating a multiple-input multiple-output (MIMO) antenna is diﬃcult since its radiation patterns and eﬃciency
are not direct indicators of MIMO communication performance [1], [2]. Especially for the terminal MIMO antennas, many antennas need to be built in a small chassis, andthis yields several problems with the MIMO communication performance. One problem is strong mutual coupling
among antenna elements. The transmitted signal from oneof the antennas is captured by other antenna elements andthis cause degradation in signal-to-noise ratio (SNR). Indeed, this phenomenon is categorized as the miss-match loss
since both the reflected and captured powers go back to thesources. On the basis of the reciprocity of the passive circuit system, this miss-match loss aﬀects the reception performance of the antenna as well. The mutual coupling also
causes distortion in the radiation pattern. Even though thisaﬀects the spatial correlation coeﬃcient, the spatial correlation is sometimes improved by mutual coupling [1]. As
described here, the mutual coupling plays an important rolein determining the channel capacity since it strongly aﬀects
these two factors, i.e. SNR and spatial correlation.
The spatial correlation characteristics of a MIMO system are determined by both propagation characteristics andManuscript received April 6, 2015.
Manuscript revised August 17, 2015.
†
The authors are with Iwate University, Morioka-shi, 020-8551
Japan.
††
The author is with Graduate School of Science and Engineering, National Defense Academy, Yokosuka-shi, 239-8686 Japan.
†††
The author is with Panasonic System Networks Co., Ltd.,
Yokohama-shi, 224-8539 Japan.
††††
The author is with Toyama University, Toyama-shi, 930-8555
Japan.
a) E-mail: honma@m.ieice.org
DOI: 10.1587/transcom.2015ISP0023

complex radiation patterns [3], [4]. The propagation characteristics depend on the various factors, e.g. Rician factor,
path distributions, and cross-polarization discrimination in
the path. Furthermore, it is significantly aﬀected by the orientation and location of the antennas. Therefore, the deterministic evaluation of the MIMO antenna is quite diﬃcult.
On the other hand, the diversity-antenna evaluation method
using the simplified path distribution model and complex
radiation patterns has been studied [5]–[7]. However, thesemethods require the 3D-complex radiation patterns for all
antenna elements. Although various methods for measuring 3D-complex radiation pattern have been used [8], [9],further simplified and cost eﬀective method is desired.
The antenna characteristics can be recognized to be
deterministic since this is determined when the antenna is
manufactured if we neglect the eﬀect of the surrounding objects such as the human body. On the other hand, the channel
always varies since the orientation and location of the antenna is uncertain in mobile communication system, such as
cellular terminals. Therefore, the MIMO antenna evaluation
method without depending on the channel characteristics isneeded if our interest is limited to the antenna performance.
S. Blanch et al. have proposed a simple method for calculating envelope correlation coeﬃcient only from S -parametersof the multiple antenna system [10], where the channel issimplified to three-dimensional (3D) uniform. If the ohmic
loss in the antennas is negligible, this is quite convenient
for evaluating MIMO antenna performance because we do
not need to consider the either the propagation environment
or complex radiation patterns. However, this method only
calculates the correlation coeﬃcients and does not directlytranslate the antenna characteristics to MIMO channel capacity. For this reason, the capacity-based evaluation of the
MIMO antennas is used in many studies [11]–[13].
By extending the idea of [10], the authors have presented a closed-form equation for calculating mean channel
capacity of MIMO antenna, where only S -parameters and
SNR are needed [14], [15]. Thanks to its closed-form expression, Monte Carlo simulations are not needed for evaluating mean channel capacity. Even though this equation
is simple and convenient, the accuracy is poor especially at
low SNR cases.
In this paper, a fairly accurate channel capacity equation only based on S -parameters is proposed. In thisequation, the eigenvalues of the S -parameter matrix of the
MIMO antenna is used. Also, mean eigenvalues of iid (inde-c 2016 The Institute of Electronics, Information and Communication Engineers
Copyright IEICE TRANS. COMMUN., VOL.E99–B, NO.1 JANUARY 201696

pendently and identically distributed) random channel matrices are used. Since the combination of the antennas andpropagation characteristics determines the channel capacity,
the interaction of them causes uncertainness in capacity calculation. To resolve this problem, the upper and lower capacity bounds are calculated from these eigenvalues, and theaverage of these two bounds is approximated as the meanchannel capacity. This means mean channel capacity can becalculated without Monte Carlo simulations. In the following part of the paper, the derivation of the equation is described. Furthermore, the validity of the proposed equation
is evaluated numerically and experimentally.
2. System Model and Proposed Equation
2.1 System Model
Figure 1 shows a conceptual sketch of the multiple antennas
and channel model. This model assumes Rayleigh channel, where suﬃciently many paths exist around the antennas, and the paths distribute in 3D random directions (3D
uniform) [2], [16]. Also, the transmitting antennas are notshown in this model because our interest here is only a setof MIMO array. This model assumes the ideally uncorrelated transmitting antenna array that has antenna elements
as many as that of the receiving array. The S -parameter matrix of M element MIMO array is defined as SA (∈ C M×M ).
This study assumes the MIMO array is lossless, i.e. it may
have the mismatch loss and mutual couplings but does nothave ohmic loss because this paper uses the approximationof [10] for calculating the correlation matrix.
2.2 Mean Channel Capacity Based on Eigenvalues of S Parameter
The 3D complex radiation pattern of i-th antenna is definedas,Di (θ, φ) = [Dθi (θ, φ), Dφi (θ, φ)]T .

When the antenna array is lossless, (2) can be simply calculated by
RA = I − SH
A SA

(3)

where I is identity matrix [10].
In Kronecker model, the MIMO channel for this
MIMO array can be generated by
H = R1/2
A HW .

(4)

HW is M × M iid channel matrix, where the its elements are
zero-mean complex Gaussian random. Figure 2 represents
the block diagram of the channel expressed by (4). Because
the ideally uncorrelated array is assumed at the transmitterside, the correlation matrix for the transmitting array is expressed by M × M identity matrix, I. For this antenna system, the instantaneous MIMO channel capacity is expressedby,
C = log2 HH Hγ+IM

H 1/2H= log2 R1/2
A HW HW RA

(5)
γ
+I
M

where γ is SNR. Note that the γ is not given value but theparameter we determine. The definition of γ in this paper isthe ratio of the signal to the noise when the lossless omnidirectional antennas are used at both the transmitting andreceiving sides. This means the actual SNR is diﬀerent fromthe determined γ because of the mismatch loss, and so on.
Since all of the matrices in (6) are M × M square, (6) can be
re-written as,
H
C = log2 RA HW HW

γ
+I
M

H
= log2 U A DA U H
A U W DW U W

(7)
γ
+I
M

Conceptual sketch of antennas and channel model.

(8)

where DA and DW are the diagonal eigenvalue matrices(1)

The correlation matrix of the MIMO array is calculated by
the radiation integral of the product of the complex patterns
as,
⎤
⎡ H
H
  ⎢⎢⎢ D1 D1 · · · D1 D M ⎥⎥⎥
1
⎥⎥⎥
⎢⎢⎢ .
..
..
RA =
(2)
⎢⎢⎢ ..
⎥⎥⎥ dΩ.
.
.
4π
⎦
⎣ H
H
D M D1 · · · D M D M

Fig. 1(6)

Fig. 2Block diagram of channel model.

HONMA et al.: CHANNEL CAPACITY EVALUATION OF MIMO ANTENNA BASED ON EIGENVALUES OF S -PARAMETER

97

for the correlation matrices calculated from antenna S parameter and iid channel matrices, respectively, and they
are represented by DA = diag[λA1 , . . . , λAM ] and DW =diag[λW1 , . . . , λW M ]. U A and UW are the eigenvector matrices corresponding to DA and DW , respectively, and areunitary matrices. From (8), it can be seen that the DA is
implicitly multiplied by DW , i.e. their eigenvalues are transformed by unitary matrix. For this case, the upper and lower
boundary of (8) can be calculated by,
M

Chigh =

log2 (1 + λAi λWi γ/M)

(9)log2 (1 + λAi λW(M+1−i) γ/M)

(10)

i=1
M

Clow =
i=1

where the relation, Clow < C < Chigh , is satisfied [17].
UH
A U W in (8) is a unitary matrix representing the interaction between the modes of the antenna and propagation, andthe resulting capacity varies depending on the trial. (9)–(10) indicate the upper bound capacity is realized when the
eigenvalue orderings of the antenna and propagation modes
agree, and the lower bound is realized when the eigenvalue
orderings of them are in reverse.
In (9)–(10), the eigenvalue of the propagation mode,
λWi , is an instantaneous value, and varies depending on thetrial, too. Since the objective of this study is the estimationof the mean channel capacity without Monte Carlo simulation, the knowledge of the statistic eigenvalue distribution
is desirable rather than the instantaneous values. This study
uses mean eigenvalue distribution of the random channel,
which is numerically calculated in advance. Therefore, (9)–(10) corresponding to mean eigenvalue distributions are approximated as,
M

Chigh log2 (1 + λAi λWi γ/M).

(11)log2 (1 + λAi λW(M+1−i) γ/M).

(12)

i=1
M

Clow 
i=1

Note that the channel capacity consists of the logarithm of
the linear function of the eigenvalues. This means geometric means of the eigenvalues, λWi , will yield better accu. .
racy rather than arithmetic means ( . N1 (log x1 + log x2 · · · +log xN ) = log(x1 x2 · · · xN )1/N ). Finally, we assume the meancapacity becomes around the middle of Chigh and Clow , andis expressed as,
Clow + Chigh.
(13)
2
As described above, the mean channel capacity can be easily
calculated from S -parameters of antenna and SNR.
Figure 3 is the flowchart of the mean capacity measurement of MIMO antenna, where the scheme diﬀerencesamong the existing and proposed methods are shown. Figure 3(a) represents a direct method, where the MIMO channels of the fabricated antenna are directly measured by theCFig. 3 Flowchart comparison of the various measurement methods of
MIMO antenna capacity.

MIMO channel sounder, and this method has been well usedin many studies [11]–[13], [18], [19]. Although the useof the existing wireless systems is eﬀective in measuring
the data-rate, the results contain non-antenna oriented factors, such as other RF device characteristics, modulation
schemes, the control of the other layer in the system, andso on. In the scheme (a), the channel sounder is costly andtakes a long time for measuring many channels because theantennas need to be placed at the various locations and orientations. Figure 3(b) shows the hybrid method, where only
the antennas are fabricated and channels are computationally generated [2], [16], [20]. This scheme does not needeither the expensive channel sounder or exhausting channel
measurements. Nevertheless, the MIMO channel needs tobe adequately modeled for evaluating the MIMO antennas.
Furthermore, the mean channel capacity cannot be straightforwardly calculated even though the antenna characteristicsof the MIMO antenna are constant and deterministic. Therefore, we still need several steps for obtaining the mean channel capacity. Figure 3(c) shows the scheme of the proposed
approach. It is obvious that the proposed method needs only
a few steps if our interest is the mean channel capacity. Although this idea has been already presented by the authors[15], the previous method has a problem in accuracy. If
the accuracy is significantly improved, the proposed method
will be more practical than the previous method.
3. Simulation
Figure 4 shows the simulated MIMO array model, whereM-element uniform linear dipole array is assumed, and d
is inter-element spacing. The S -parameter matrix and 3D
complex radiation patterns of the MIMO array are calculated by Moment Method. Here, each radiation pattern corresponding to each feed port is calculated by terminatingother ports with 50 Ω loads. In order to evaluate the accuracy of the proposed method, two types of Monte Carlo simulations are used. First one uses Kronecker model, which isdefined by (4). This model uses Gaussian random matri-

IEICE TRANS. COMMUN., VOL.E99–B, NO.1 JANUARY 201698

Fig. 4

Simulated antenna model.

ces, where each of their elements has zero-mean complex
Gaussian distribution with σ2 = 1 (σ2 : variance). For thisscheme, the correlation matrix calculated from S -parameter
matrix of the antenna is used. This means, the error causedby the ohmic loss cannot be evaluated correctly. Second
one is a geometry-based channel model (GCM) [16], whichuses complex radiation patterns and geometrically definedrays that go through the randomly distributed scatterers at
the transmitting and receiving sides. The distribution of
the instantaneous amplitude of the channel corresponds toGaussian distribution, and the variance of the channel is normalized to 1 when the omni-directional antenna is used atboth the transmitting and receiving sides. That is, the entire
radiation characteristics including the eﬃciency of the antennas are taken into account, whereas the Kronecker model
described above cannot consider the ohmic loss. Also, 3Duniform path distribution with no direct path is assumed,
and the number of the paths is set to 100. To uncorrelatedtransmitting side, the ideal omni-directional antennas with
suﬃciently wide inter-element spacing (10 wavelengths) areused. For both Monte Carlo simulations, the number of thesimulation trials is set to 1000.
In the following discussion, the Kronecker model isused only for the verification of the intermediate results ofthe proposed method. Also, it must be noted that theseMonte Carlo simulations are just for the verification of the
proposed method, and not used in the proposed method.
To verify the assumption that the mean capacity is approximated by (13), the upper and lower bounds calculatedby (9) and (10) are evaluated by the Monte Carlo simulation based on the Kronecker model. Figure 5 shows cumulative distribution function (CDF) of Chigh and Clow , wherethe values are normalized by the instantaneous capacity, C,
i.e. straightforwardly calculated by (5). In this simulation,
the number of the antenna elements is set to M = 4, and
SNR is 20 dB. The element spacings for Fig. 5(a) and (b)are d = 0.2λ0 , 1.0λ0 (λ0 : wavelength in vacuum), respectively. From this result, it can be seen that C is just middleof the distributions of the upper and lower bounds. This result supports (13) well approximate mean channel capacity.
Also, the result for d = 1.0λ0 indicates that Chigh and Clow
are quite close. In this case, the antennas are suﬃciently
separated, and the diﬀerence in λA1 ∼ λAM is small. Thismeans (9) and (10) yield almost similar values.
Figure 6 shows E[Chigh ] and E[Clow ] versus d, where
the Monte Carlo simulation based on the Kronecker model
was used. (E[Chigh ] + E[Clow ])/2 and actual mean capacity,
E[C], calculated by (5) are also shown. Where, the operation E[·] represents ensemble average. It can be seen that

Fig. 5 CDF of capacity upper bound (Chigh ) and lower bound (Clow ) for
M = 4.

Fig. 6

Mean channel capacity versus element spacing, d.

E[C] is in almost middle of E[Clow ] and E[Chigh ]. Similarly to Fig. 5, E[Chigh ] and E[Clow ] approach when the antenna spacing, d, becomes large. Also, it can be seen that

HONMA et al.: CHANNEL CAPACITY EVALUATION OF MIMO ANTENNA BASED ON EIGENVALUES OF S -PARAMETER

99
Table 1

Eigenvalue table: geometric means of eigenvalues (λWi ).

aaaa Number ofM
aantennas,
Index
aof eigenvalues, i aa
a
1
2
3
4
5
6
7
8

2

3

4

5

6

7

8

3.05
0.28
–
–
–
–
–
–

6.11
1.89
0.19
–
–
–
–
–

9.39
4.16
1.38
0.14
–
–
–
–

12.79
6.77
3.21
1.09
0.11
–
–
–

16.28
9.59
5.40
2.62
0.91
0.09
–
–

19.84
12.56
7.84
4.52
2.23
0.78
0.08
–

23.42
15.64
10.45
6.68
3.90
1.94
0.68
0.07

Fig. 7 Arithmetic and geometric means of eigenvalues in Gaussian random matrix.

E[Chigh ] and E[Clow ] approach when SNR is high. E[C] and(E[Chigh ] + E[Clow ])/2 well agree even when d and SNR are
varied, and this supports the validity of the approximation
E[C]  (E[Chigh ] + E[Clow ])/2.
The discussion hereinbefore in this section was basedon the Monte Carlo simulation. The important objective of
this work is evaluating the mean capacity without Monte
Carlo simulation. As mentioned in 2.2, the mean eigenvalues of the random matrix are used for calculating meancapacity. Figure 7 shows arithmetic and geometric meansof the eigenvalues, where all of the elements of the randommatrix have a Gaussian distribution with zero-mean and unit
variance, and mean values are calculated by 106 trials. Notethat the index number of the eigenvalue is given in descending order of eigenvalue magnitude. It can be seen that thegeometric mean value is always smaller than the arithmetic
mean, and it is well known as the inequality of arithmetic
and geometric means.
In Table 1, the geometric means of the eigenvalues arelisted, where they are exact identical to the values shown in
Fig 7. This table can be used for calculating the capacity by(11)–(13), and Monte Carlo simulation is no longer needed.
Figure 8 shows the relationship between d and capacity,which is calculated by the proposed equations, (12)–(13),where the number of the elements is M = 2. For comparison, the capacity, which is calculated by the Monte Carlosimulation, is shown, too. In the remaining part of the paper, the Monte Carlo simulations are based on GCM. Even
though the proposed equation with geometric means of theeigenvalues is slightly lower than the GCM for low SNR

Fig. 8Capacity versus element spacing, d (M = 2).

case, they well agree for high SNR case. Furthermore, it isfound that the results with arithmetic means of the eigenvalues suﬀer from large errors in capacity.
Figure 9 also shows the relationship between d and capacity when the number of element is M = 4. From thisresult, it can be seen that the capacity equation using arithmetic means of the eigenvalues always overestimates the capacity, and that using geometric ones yields less error regardless of SNR. From the results shown in Figs. 8 and 9, it
is found that the proposed equation with geometrical meansof the eigenvalues underestimates the capacity only when
S NR = 0 dB and M = 2, whereas all other results overestimate the capacity. Although the reason of this changeoveris unclear, this is suspected due to the approximations in(11)–(13). Figure 10 shows the relationship between capac-

IEICE TRANS. COMMUN., VOL.E99–B, NO.1 JANUARY 2016100

Fig. 9

Capacity versus element spacing, d (M = 4).
Fig. 10

Capacity versus SNR.

ity and SNR for the cases, M = 2 and 4. ‘Conventional’
represents the capacity calculated by the equation shown in[15], and it suﬀers from underestimation at low SNR regionand overestimation at high SNR region. This is due to thehigh SNR approximation. Also, it is found that the conventional equation causes large error compared to GCM evenin high SNR case. This error comes from the inequality ofarithmetic and geometric means [15]. On the other hand, theproposed equation with geometric means of the eigenvalues
oﬀers highly accurate capacity independently of the numberof the antenna elements, spacing, and SNR.
4. Measurement
The validity of the equation is experimentally assessed inthis section. Figure 11(a) and (b) are photos of tested MIMO
arrays. The monopole antennas are used for this experimentbecause the conducting loss is relatively small and the feed
cables little aﬀect the radiation characteristics thanks to theground plane. Two and four element arrays are tested, wherea square arrangement is used for four element array. The
length of the monopole is around quarter wavelength in vacuum and the size of the ground plane is 1.62λ0 × 1.62λ0
for both arrays. The operation frequency for both arrays is
2.4 GHz.
Figure 12(a) and (b) show the capacity as a function ofantenna spacing, d. The capacity of the proposed equation
is calculated using S -parameter matrix measured by vector

Fig. 11

Measured monopole array.

network analyzer. The results shown by ‘GCM’ are calculated by geometry-based channel model using measured 3D
complex radiation patterns, and all of the channel parameters are identical to that in Sect. 3. It can be seen that almost
constant channel capacity is obtained for M = 2 and thistendency agrees with that in Fig. 8 (d > 0.35λ0 ). This isHONMA et al.: CHANNEL CAPACITY EVALUATION OF MIMO ANTENNA BASED ON EIGENVALUES OF S -PARAMETER

101

Fig. 13

Capacity versus SNR (measurement).

Also, the equation uses mean eigenvalue distribution of random iid channel, which can be calculated in advance of theevaluation. The proposed equation is valid only when thefollowing conditions are assumed:- same number of antennas at both transmitting and receiving sides- no ohmic loss
- 3D uniform Rayleigh environment

Fig. 12

Capacity versus d (measurement).

because the mutual coupling is too small to aﬀect the capacity for this case. On the other hand, the capacity for
M = 4 is degraded when d is small because the square arrayyields mutual coupling higher than linear 2 element array(M = 2) does. It can be seen that the proposed method
slightly overestimates the channel capacity. To identify thereason of this discrepancy, the radiation eﬃciency is evaluated using 3D radiation patterns of the monopole array,and it is found that the radiation eﬃciency of tested arrayis about 93 ∼ 99%. Therefore, it is considered the ohmic
loss causes error in this equation because 100% radiationeﬃciency is assumed for derivation. Nevertheless, the channel capacity is estimated within 12% error even for low SNRcase (SNR=10 dB). Since the error is less than 9% for high
SNR case (SNR = 30 dB), the accuracy of this equation isabout 9% ∼ 12% for this experiment.
Figure 13 shows the relationship between the capacity and SNR. It can be seen that the results with the proposed equation well agree with the GCM results regardlessof SNR. From this result it is found that the proposed equation well estimates the capacity even when SNR is low.
5. Conclusion
This paper has proposed an equation for mean channel capacity that uses only the S -parameter and SNR. This method
uses the eigenvalue table of the correlation matrix, which is
calculated only from S -parameter matrix of MIMO antenna.

The accuracy of the proposed equation was evaluated numerically and experimentally. From the numerical analysis, it is found that the geometric means of eigenvalues give
more accurate results than the arithmetic ones. The experiment results showed that the proposed equation oﬀers
the mean channel capacity within 12% error regardless of
the SNR. These results indicate that the proposed equationis quite convenient to calculate the mean channel capacity
since it does not need Monte Carlo simulations of MIMO
channel, which has prevented the direct evaluation of MIMO
antennas.
Acknowledgements
This research was partially supported by JSPS KAKENHI(25709030).
References[1] J.W. Wallace and M.A. Jensen, “Mutual coupling in MIMO wirelesssystems: A rigorous network theory analysis,” IEEE Trans. WirelessCommun., vol.3, no.4, pp.1317–1325, July 2004.
[2] B.K. Lau, J.B. Andersen, G. Kristensson, and A.F. Molisch, “Impact of matching network on bandwidth of compact antenna arrays,”
IEEE Trans. Antennas Propag., vol.54, no.11, pp.3225–3238, Nov.
2006.
[3] D.W. Browne, M. Manteghi, M.P. Fitz, and Y. Rahmat-Samii,“Experiments with compact antenna arrays for MIMO radio
communications,” IEEE Trans. Antennas Propag., vol.54, no.11,
pp.3239–3250, Nov. 2006.
[4] P.-S. Kildal and K. Rosengren, “Correlation and capacity of MIMO
systems and mutual coupling, radiation eﬃciency, and diversity gainof their antennas: Simulations and measurements in a reverberationchamber,” IEEE Commun. Mag., vol.42, no.12, pp.104–112, Dec.
2004.
[5] T. Taga, “Analysis for mean eﬀective gain of mobile antennas in

IEICE TRANS. COMMUN., VOL.E99–B, NO.1 JANUARY 2016

102

land mobile radio environments,” IEEE Trans. Veh. Technol., vol.39,
no.2, pp.117–131, May 1990.
[6] K. Ogawa and T. Matsuyoshi, “An analysis of the performance ofa handset diversity antenna influenced by head, hand, and shoulder eﬀects at 900 MHz Part I — Eﬀective gain characteristics,” IEEE
Trans. Veh. Technol., vol.50, no.3, pp.830–844, May 2001.
[7] K. Ogawa, T. Matsuyoshi, and K. Monma, “An analysis of the performance of a handset diversity antenna influenced by head, hand,and shoulder eﬀects at 900 MHz Part II — Correlation characteristics,” IEEE Trans. Veh. Technol., vol.50, no.3, pp.845–853, May
2001.
[8] Y. Okano and K. Cho, “Antenna measurement system for mobile
terminals,” NTT DoCoMo Technical J., vol.9, no.2, pp.43–50, 2007.
[9] L. Duchesne, P. Garreau, N. Robic, A. Gandois, P.O. Iversen, and G.
Barone, “Compact multi-probe antenna test station for rapid testingof antennas and wireless terminals,” Technical Seminar on Antenna
Measurements and SAR (AMS 2004), pp.101–105, Jan. 2004.
[10] S. Blanch, J. Romeu, and I. Corbella, “Exact representation of antenna system diversity performance from input parameter description,” Electron. Lett., vol.39, no.9, pp.705–707, May 2003.
[11] J.W. Wallace, M.A. Jensen, A.L. Swindlehurst, and B.D. Jeﬀs, “Experimental characterization of the MIMO wireless channel: Data acquisition and analysis,” IEEE Trans. Wireless Commun., vol.2, no.2,
pp.335–343, March 2003.
[12] D.W. Browne, M. Manteghi, M.P. Fitz, and Y. Rahmat-Samii,“Experiments with compact antenna arrays for MIMO radio
communications,” IEEE Trans. Antennas Propag., vol.54, no.11,
pp.3239–3250, Nov. 2006.
[13] N. Honma, K. Nishimori, R. Kudo, Y. Takatori, T. Hiraguri, and
M. Mizoguchi, “A stochastic approach to design MIMO antenna
with parasitic elements based on propagation characteristics,” IEICE
Trans. Commun., vol.E93-B, no.10, pp.2578–2585, Oct. 2010.
[14] N. Honma, Y. Tsunekawa, and K. Ogawa, “Evaluation of capacity
equation only based on S-parameter for realistic MIMO antenna,”2013 IEEE Antennas and Propagation Society International Symposium (APSURSI), pp.548–549, 2013.
[15] N. Honma, H. Sato, K. Ogawa, and Y. Tsunekawa, “Accuracyof MIMO channel capacity equation based only on S -parametersof MIMO antenna,” Antennas Wirel. Propag. Lett., vol.14,
pp.1250–1253, June 2015.
[16] A.F. Molisch, “A generic model for MIMO wireless propagation
channels in macro- and microcells,” IEEE Trans. Signal Process.,
vol.52, no.1, pp.61–71, Jan. 2004.
[17] M. Fiedler, “Bounds for the determinant of the sum of hermitianmatrices,” Proc. Amer. Math. Soc., vol.30, no.1, pp.27–27, 1971.
[18] J.P. Kermoal, L. Schumacher, K.I. Pedersen, P.E. Mogensen, and
F. Frederiksen, “A stochastic MIMO radio channel model with experimental validation,” IEEE J. Sel. Areas. Commun., vol.20, no.6,
pp.1211–1226, Aug. 2002.
[19] V. Jungnickel, V. Pohl, and C.V. Helmolt, “Capacity of MIMO systems with closely spaced antennas,” IEEE Commun. Lett., vol.7,
no.8, pp.361–363, Aug. 2003.
[20] Y. Okano and K. Cho, “Monopole antenna array arrangement forcard-type mobile terminal,” Proc. 2004 IEEE Radio and WirelessConference, pp.415–418, 2004.

1/2 H
R1/2
A = U A DA U A ,(A· 2)

because
1/2
1/2 H
1/2 H
R1/2
A RA = U A DA U A U A DA U A= U A DA U HAIn this appendix, the derivations of (7) and (8) from (6) aredetailed. The eigenvalue decomposition of the correlation
matrix of (3) is expressed as,(A· 1)

where U A (∈ C M×M ) and DA (∈ C M×M ) are the eigenvector(A· 3)(A· 4)(A· 5)= RA .
Also,
1/2 H H
1/2 H
R1/2H
R1/2
A
A = (U A DA U A ) U A DA U A= U A DA U H
A(A· 6)(A· 7)(A· 8)= RA

is satisfied. These relations are used in the following part ofthe derivation.
Now, the part of the determinant in (6) is considered
for simplicity, and can be rewritten as,
H 1/2HR1/2
A HW HW RA

γ
+I
M

γ
+ R−1/2
R−1/2H
R1/2H
A
A
A
M
H γ
+ R−1
= R1/2
HW HW
R1/2H
A
A
A
M
H γ
+I
= R1/2
R1/2H
R−1
RA HW HW
A
A
A
M
H γ
+I .
= RA HW HWM
H= R1/2HW HW
A(A· 9)As shown the derivation above, (A· 9) validates the transforH
is replaced by its
mation from (6) to (7). Finally, HW HWeigenvalue decomposition:
H
H
HW HW
= U W DW U W,(A· 10)where UW (∈ C M×M ) and DW (∈ C M×M ) are the eigenvector
matrix and diagonal eigenvalue matrix, respectively, and we
obtainH
RA HW HWγ+I
M

H= U A DA U H
A U W DW U W

γ
+I ,
M

which supports the validity of (8).

Appendix: Derivations of (7) and (8)RA = U A DA U HA,matrix and diagonal eigenvalue matrix, respectively. Since
RA is Hermitian matrix, all eigenvalues are real. The eigenvalues are positive or zero because they represent the reception power corresponding to the eigenmodes. The squareroot of (A· 1) is,(A· 11)HONMA et al.: CHANNEL CAPACITY EVALUATION OF MIMO ANTENNA BASED ON EIGENVALUES OF S -PARAMETER

103

Naoki Honma
received the B.E., M.E., andPh.D. degrees in electrical engineering from Tohoku University, Sendai, Japan in 1996, 1998,
and 2005, respectively. In 1998, he joined the
NTT Radio Communication Systems Laboratories, Nippon Telegraph and Telephone Corporation (NTT), in Japan. He is now working for
Iwate University. He received the Young Engineers Award from the IEICE of Japan in 2003,
the APMC Best Paper Award in 2003, the Best
Paper Award of IEICE Communication Society
in 2006, and 2014 Asia-Pacific Microwave Conference Prize in 2014, respectively. His current research interest is MIMO system and its applications. He is a member of IEEE.

Kentaro Murata
received the B.E. and M.E.
degrees in electrical and electronics engineeringfrom Iwate University, Morioka, Japan in 2011
and 2014, respectively. He is currently working
toward the Ph.D. degree with Graduate Schoolof Science and Engineering, National Defense
Academy of Japan, Yokosuka, Japan. His current research interests include feeding methods
for multiple-input-multiple-output antenna systems.

Hiroshi Satowas born in Tokyo, Japan, on
August 2, 1975. He received B.S. and M.S. degree in electrical engineering from Tokyo City
University, Japan, in 1998 and 2000, respectively. He is currently researching for his Ph.D.
degree in Graduate School of Engineering from
Chiba University, Japan. From 2004 to 2012,he has been with Panasonic Mobile Communication Co., Ltd., Yokosuka, Japan. And is currently a leader of the R&D project for mobile
phone antennas in Panasonic System Networks
Co., Ltd, Yokohama, Japan. His research interests include small antenna,
MIMO antenna and decoupling technique. He was the recipient of the Best
Paper Award from the Institute of Electronics, Information and Communication Engineers (IEICE) Transactions of Japan in 2012. He is a memberof IEEE.

Koichi Ogawa
was born in Kyoto on May
28, 1955. He received his B.S. and M.S. degrees in electrical engineering from Shizuoka
University in 1979 and 1981, respectively. He
received the Ph.D. degree in electrical engineering from the Tokyo Institute of Technology, Tokyo, Japan, in 2000. He joined Matsushita Electric Industrial Co., Ltd., Osaka, Japan, in 1981.
He is currently a Professor with Toyama University, Toyama, Japan. His research interests include compact antennas, diversity, adaptive, andMIMO antennas for mobile communication systems, and electromagnetic
interaction between antennas and the human body. His research also includes millimeter-wave circuitry and other related areas of radio propagation. Dr. Ogawa was the recipient of the OHM Technology Award fromthe Promotion Foundation for Electrical Science and Engineering in 1990,
based on his accomplishments and contributions to the millimeter-wave
technologies. He was also the recipient of the TELECOM System Technology Award from the Telecommunications Advancement Foundation (TAF)in 2001, based on his accomplishments and contributions to portable handset antenna technologies, and the Best Paper Award from the Institute ofElectronics, Information and Communication Engineers (IEICE) Transactions of Japan in 2009 and 2012. He is a Senior Member of the IEEE andis listed in Who’s Who in the World. He is currently the Chair of the IEEE
AP-S Nagoya Chapter.

Yoshitaka Tsunekawa
received the B.E.
degree from Iwate University, Morioka, Japan,
in 1980 and the M.E. and the Doctor of Engineering degrees from Tohoku University, Sendai, Japan, in 1983 and 1993, respectively. Since
1983, he has been with the Faculty of Engineering, Iwate University, where he is now a professor at the Department of Electrical Engineeringand Computer Science. His current research interests include digital signal processing and digital control. He is a member of the Instituteof Electronics, Information and Communication Engineers of Japan, andIEEE.