IEICE TRANS. COMMUN., VOL.E98–B, NO.11 NOVEMBER 2015

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PAPER

Fast MIMO Channel Calculation Technique for Multi-Antenna
System Using Signal Modulation for FDTD Method
Kazuma OUCHIDA† , Student Member, Naoki HONMA†a) , and Yoshitaka TSUNEKAWA† , Members

SUMMARY This paper proposes a new method that combines signal
modulation and FDTD (Finite-Diﬀerence Time-Domain) simulations to reduce the computation time in multiple-antenna analysis. In this method,signals are modulated so as to maintain orthogonality among the excited
signals; multiple antennas are excited at the same time. This means justone FDTD simulation is needed whereas the conventional method demands
as many simulations as there are transmitting antennas. The simulation of
a 2×2 multi-antenna system shows that the proposed method matches theperformance of the conventional method even though its computation time
is much shorter.
key words: FDTD, signal modulation, channel capacity, multi-antenna,
MIMO

1. Introduction
Technologies for enhancing the data rate of wireless communication are being actively pursued, and diversity antenna[1] and Multiple-Input Multiple-Output (MIMO) systems[2] are seen as key advances. The characteristics of the multiple antennas greatly impact the diversity gain or channel
capacity. The eﬀect of the antenna characteristics on the
MIMO channel was generalized by Wallace et al. [3], but
the performance of a MIMO system strongly depends on
the interaction between the antenna system and propagation
environment. Therefore, evaluating a multiple antenna system still demands extensive simulations to fully elucidate
the interaction. This incurs very high computation cost evenwith the recent progress in computers. Hence, an eﬃcientmethod of MIMO channel simulation is needed.
The ray-tracing method [4]–[6] and the FDTD (FiniteDiﬀerence Time-Domain) method [7], [8] are currently being used to evaluate propagation characteristics. Since the
ray-tracing method is based on geometric optics theory,
highly complex structures explode the computation timeand degrade analysis accuracy. The FDTD method, on theother hand, divides the analysis region into small cells, andMaxwell’s equations are applied to explain the electromagnetic interactions among the cells. The FDTD method isconvenient for analyzing detailed structures including theantennas and propagation environments. For this reason,
many large scale analyses have been based on the FDTD
method [8]. Unfortunately, its calculation time and memory
requirement increase in proportion to the total volume of theManuscript received February 23, 2015.
Manuscript revised June 10, 2015.
†
The authors are with the Graduate School of Engineering,
Iwate University, Morioka-shi, 020-8551 Japan.
a) E-mail: honma@m.ieice.org
DOI: 10.1587/transcom.E98.B.2300

analysis space.
Fast FDTD analysis methods such as ADI (AlternatingDirection Implicit) method [9] and sub-cell method [10]have been proposed. In the ADI-FDTD method, the time
step is not restricted by the CFL (Courant-Friedrichs-Lewy)
condition. This method can reduce the computation time bysetting large time steps as appropriate. On the other hand,the sub-cell method can deal with geometries that do notsuit the cells used. Since there is no need to use fine cells
throughout the entire analysis region, it can reduce bothcomputation memory and time. Even though these techniques are eﬀective in alleviating the computation load, analyzing an array antenna with a voluminous propagation environment needs long computation time that is proportion to
the number of the antennas since each antenna needs to be
excited individually. For this reason, a fast FDTD analysis
suitable for multiple-antenna systems is required.
A fast analysis method for multiple-antenna systems
using OFDM (Orthogonal Frequency Division Multiplexing) has been proposed [11]. This method enables the simultaneous analysis of multiple antennas since the excitation signals are spectrally orthogonalized by exploiting
OFDM modulation. However, this method cannot analyze
two channels with the same frequency and so the frequency
response is intermittent.
To resolve this drawback, we proposed a fast analysis
method using the spread spectrum technique [12], [13]. Thismethod enables the simultaneous analysis of multiple antennas since the excitation signals are orthogonalized by exploiting non-identical spreading codes. However, the wideband signals generated by spreading codes average the frequency characteristics of the channel, and the accurate channel response at the specific frequency cannot be observed.
Otherwise, a long period of the signal is needed for a highspectral resolution [14], but this causes serious increase in
the FDTD time step. Therefore, neither this method nor [11]can analyze the MIMO channel in a multipath environment.
In this paper, we propose a fast analysis method for themulti-path environment. Its key advance is to introduce signal modulation to FDTD analysis. The modulated signals
for multiple transmitting antennas are made orthogonal toeach other, so the signals can be well separated (isolation ishigh) at the receiving antennas. That is, the multiple transmitting antennas are excited simultaneously and the MIMO
channel can be calculated with just one simulation. The
remainder of this paper describe the proposed scheme andmechanism in detail. Numerical analysis results are also de-c 2015 The Institute of Electronics, Information and Communication Engineers
Copyright OUCHIDA et al.: FAST MIMO CHANNEL CALCULATION TECHNIQUE FOR MULTI-ANTENNA SYSTEM USING SIGNAL MODULATION FOR FDTD METHOD

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scribed; they show that the proposed method oﬀers much
faster analysis than the FDTD method with the conventional
Gaussian pulse but with slight loss of accuracy.
2. The Concept of the Proposed Method
Figure 1 shows the concept of the proposed FDTD analysis
method. In the FDTD method, the propagation channel iscalculated using the voltage and current at the port of thereceiving antenna. The total voltage and current in the time
domain can be calculated first. After that, the voltage andcurrent waveform in the time domain are transformed intofrequency domain values by Fourier transformation. The
channel is obtained from the voltage and current in the frequency domain. Figure 2 shows the transmitting and receiving ports. The transmitted signal vector and the received
signal vector in the frequency domain are expressed as,
1(1)
x = (uT /Z0 1/2 + iT Z0 1/2 )2
1(2)
y = (uR /Z0 1/2 − iR Z0 1/2 ),2where Z0 is the reference impedance used to define thescattering parameters. uT = [vT 1 , ..., vT Mt ] and uR =[vR1 , ..., vRMr ] are voltage vectors in the frequency domain
at the port of the transmitting and receiving antenna, respectively. iT = [iT 1 , ..., iT Mt ] and iR = [iR1 , ..., iRMr ] are current

Fig. 1

Concept of the proposed method.

Fig. 2

Transmitting and receiving ports.

vectors in the frequency domain at the port of the transmitting and receiving antenna, respectively. Next, each of
the transmitted signals is defined by the plurality of symbols V = [uT 1 , ..., uT N ]. The transmitted signal vector is
calculated by (1), and the transmitted signal matrix X =[x1 , ..., xN ] is generated. The number of transmitted symbols, N, is made equal to the number of transmitting antennas. The transmitted symbol length is T [time step]. Thenin the time domain, guard times, Gt , are inserted between
each of the transmitted symbols to prevent inter-symbol interference at the receiver due to multi-path delay spread in
the radio channel. All antennas are excited at the same timeby these signals. On the receiving side, the received signal
vector is calculated by (2), and the received signal matrix
Y = [y1 , ..., yN ] is generated. The MIMO channel is calculated by
H = YX−1 .

(3)

3. Numerical Results
3.1 Analysis Condition
The numerical analysis model of this study assumes dipole
arrays, see Fig. 3. In order to evaluate the MIMO channel in a multi-path environment, the dipole arrays are surrounded by concrete walls, one of which is pierced by alarge window. Two types of multi-path environments, i.e.
line-of-sight (LOS) and non-line-of-sight (NLOS), are evaluated in this study. Both models are identical except for

Fig. 3

Analysis model.

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Table 1

Parameters of the structures.
Wall
Window
Relative permittivity r
5.5 [15]
7.2
Relative permeability μr
1
1
Electrical conductivity σ
0.023
0.0293Thickness
60 mm
10 mm

Table 2

Parameters for analysis.

Antenna
Feeding model
Cell size
Absorbing B.C.
Number of PML layer
Analysis frequency
Time step

Dipole antenna
Delta-gap feed
10 mm
PML
8
1.125 GHz
19.26 psa partition wall between transmitting and receiving arrays.
The partition wall is located at the center of the room toblock the direct path between transmitting and receiving arrays for NLOS model. Parameters of the structures are summarized in Table 1. The distance between the transmitting
and receiving antennas is D1 [m], and element spacing is
D2 [m]. The distance between each element and the wallis 0.5 m. Details of the FDTD analysis configurations are
summarized in Table 2. A Gaussian pulse and modulated
signal pulse are used as the excitation signal. The latter isgenerated by BPSK (Binary Phase Shift Keying), and theexcitation signal is given by
vT lk (t) = sin 2π f t − (k − 1)(T + Gt ) + θlk ,(4)

l = 1, · · · , Mt
θlk = 0, π.
(5)
k = 1, · · · , N
θlk is determined to make the signal sets orthogonal to eachother because the orthogonal signal sets oﬀers high accuracy
in the calculation (3). In the frequency domain, the signal setof k-th period can be expressed as,⎞⎛⎞⎛⎜⎜⎜ exp( jθ1k ) ⎟⎟⎟⎜⎜⎜ x1k ⎟⎟⎟
j ⎜⎜
⎟⎟
⎟⎟⎟⎟
⎜⎜
..
xk = ⎜⎜⎜⎜ ... ⎟⎟⎟⎟ = √ ⎜⎜⎜⎜
(6)
⎟⎟⎟ .
.
⎟⎠
⎜
⎜⎝
Mt ⎝
⎠
exp( jθ Mt k )x Mt kTherefore, the signal matrix can be written as,⎞⎛⎜⎜⎜ x11 · · · x1N ⎟⎟⎟⎜⎜.. ⎟⎟⎟⎟..
X = ⎜⎜⎜⎜ ...
.
. ⎟⎟⎟
⎜⎝
⎠
x Mt 1 . . . x Mt N⎛⎞⎜⎜ exp( jθ11 ) · · · exp( jθ1N ) ⎟⎟⎟⎟⎟⎟j ⎜⎜⎜⎜
..
..
..
⎟⎟⎟ .
= √ ⎜⎜⎜
.
.
.
Mt ⎝⎜
⎠⎟
exp( jθ Mt 1 ) · · · exp( jθ Mt N )(7)

In order to realize the orthogonality among the signal sets,
the signal matrix, X, needs to be unitary. Hence, θlk is chosen to satisfy the relation, XX H = I. As an example, thesignal set for Mt = 2 is determined as,
0 0θ11 θ12=.
(8)
θ21 θ220 π

Fig. 4Modulated pulse waveforms.

Fig. 5

Received signal waveforms.

Therefore, the signal matrix is expressed as,

j 1 1
X= √
.
2 1 −1(9)which satisfies unitary condition and is used in the following
analysis.
In this study, an in-house-developed FDTD code wasused for both conventional and proposed analyses, where
basic and standard FDTD algorithm written in FORTRAN
90 was used.
3.2 Channel Characterization with the Guard Time Lengthand Symbol Length
Figure 4 shows the modulated pulse waveforms. Antenna
Tx1 transmits signal 1 shown in Fig. 4(a), and antenna Tx2transmits signal 2 shown in Fig. 4(b). Here, the symbol
length is set to T = 500 time steps, and the guard time
length, Gt , is 300 time steps. In the proposed method, analyses that use modulated pulses require 2(T +Gt ) [time steps]
for excitation and analysis. This is because guard times areinserted between each of the transmitted symbols to prevent
inter-symbol interference at the receiver that would otherwise be caused by multi-path delay spread.
Figure 5 shows received signal waveforms, which isthe simulated in NLOS model (b). The distance between thetransmitting and receiving antennas is D1 = 15 m and element spacing is D2 = 0.1 m. Antenna Rx1 received signal 1
is shown in Fig. 5(a), and antenna Rx2 received signal 2 isshown in Fig. 5(b). Nt must set so that the excitation signal
reaches the receiving antenna, i.e.,
Nt =

D1
,
vc Δt

(10)

where vc is the wave velocity in a vacuum, Δt is the time

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Fig. 6

Channel error versus symbol length.

Fig. 7Channel error versus guard time length.

Fig. 8Channel error versus element spacing.

step. Figure 5 shows that the propagation delay time is about
50 ns. This value is equal to the value calculated by (10).
Figure 6 plots channel error versus symbol length for
LOS and NLOS cases. The channel error is defined as,J=||Href − H||2F||Href ||2F

,

(11)

where H is the propagation channel to be evaluated, andHref is the propagation channel obtained by the Gaussian
pulse with suﬃciently long time analysis (10000 time steps).
In the method using a modulated signal, the number of time
steps required for the analysis is given by,
Ntotal = Nt + 2(T + Gt ),(12)

where T is the length of the excitation symbol and Gt is
the length of the guard time. In this simulation, Gt = 300
time steps. The symbol length, T , is changed from 100
to 1000 time steps. The distance between the transmitting
and receiving antennas is D1 = 15 m, and element spacing is D2 = 0.1 m. The results in Fig. 6 show that increasing the symbol length decreases the channel errors. This
also shows that the error in NLOS environment is remaining
even when the symbol length is increased whereas that in
LOS decreases with symbol length. It is considered that thepower of the direct path is dominant and the power of thedelayed multipath component is almost negligible in LOS
environment. Nevertheless, the error at T > 200 time steps
is lower than −15 dB for both environments.
Figure 7 shows channel error versus guard time length.
In this simulation, T = 256 time steps. The guard time
length, Gt , is changed from 0 to 2000 time steps. Even
though the error in NLOS environment is larger than thatof the LOS environment, the suﬃciently long guard time
oﬀers fairly accurate results. It is found that the proposed
method can analyze the MIMO channel if the guard time
length is about half the delay spread, which was about 40 ns(2000 time steps). Here, this delay spread is fairly typicalfor small indoor environment at this frequency band [16].
From the results shown here, it is found that NLOS environment yields larger error than LOS environment. Since
the accuracy for LOS is obviously higher than that of LOS,NLOS environment is evaluated in the following discussion.
3.3 Comparison with the Conventional Method
Figure 8 shows channel error versus element spacing, where
the error is the channel diﬀerence from the result with Gaussian pulse with long calculation time and is defined by (11).
The results obtained by the proposed method are comparedto the results obtained by the conventional method. Here,the number of time steps, Ntotal , is 5300. In the analysis
using the Gaussian pulse, two individual analyses for two
antennas are performed. In order to equalize the total calculation time, the number of time steps for Gaussian pulse is
set to Ntotal /2 = 2650 time steps. ‘w/o Gt ’ shows the results
for the case of not inserting a guard time. In this simulation, T = 256 and Gt = 1000 time steps. The output of
conventional method without Gaussian pulse contains large
error. The reason is the propagation time between the transmitting and receiving antennas. Since the signal transmitted
from one antenna cannot reach the other antenna, the channel cannot be calculated correctly. The results for the caseof not inserting a guard time, large error due to inter-symbolinterference is present. However, the proposed method cananalyze the two antennas at once, and the channels obtainedby the proposed method agree very well.
Figure 9 shows Signal-to-Noise Ratio (SNR) versus el-

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Fig. 9SNR versus element spacing.
Fig. 11

Fig. 10Channel capacity versus element spacing.

Spatial correlation versus element spacing.
Fig. 12

ement spacing. The SNR is given by
Tr(HH H )P
SNR = 10 log10[dB],
Mt Mr σ2(13)where, P is the transmission power and σ2 is the noise
power. In this simulation, transmission power is 0 dBm,
and noise power is −80 dBm. ‘Exact’ shows the SNR obtained by a long time (10000 time steps) calculation with theGaussian pulse. The SNR obtained by the proposed method
agrees very well with the exact result. However, it is seenthat SNR of the conventional method is about 40 dB.
Figure 10 plots spatial correlation versus element spacing. It is seen that the diﬀerence between ‘Exact’ and the
proposed method is about 0. However, large error is present
with the results corresponding to w/o Gt and Gaussian pulse.
As discussed in Fig. 8, the delayed multipath components
cause inter-symbol-interference, and the channel cannot be
estimated correctly for w/o Gt . In conventional method, thephysical time in the simulation is not suﬃcient for the signal
reaching the receiving antennas, and the channel cannot becalculated correctly, too.
Figure 11 plots channel capacity versus element spacing. The channel capacity is obtained as
H P+ I [Bits/s/Hz],(14)
C = log2 det HH
Mt σ2

Channel error versus total time steps.

where, transmission power is 0 dBm, and noise power is−80 dBm. The channel capacities obtained by the proposed
method agree very well with the exact result. However, it isseen that channel capacities of the conventional method areabout 20 bits/s/Hz.
3.4 Comparison of the Analysis Time Required
We compared the analysis time required by the proposed
method to that by the conventional method.
Figure 12 plots channel error versus total time steps.
Here, the distance between the transmitting and receiving
antennas is D1 = 15 m, and element spacing is D2 = 0.1 m.
The symbol length, T , is 256 time steps, and guard time
length, Gt , is 1000 time steps. Large error is present in theGaussian pulse results when the total time steps, Ntotal isabout 5000. On the other hand, the channel error of the proposed method is only −20 dB or so. For the channel errorof −15 dB, the proposed method needed 4305 time steps,
while the conventional method with Gaussian pulse took
5883 time steps. Therefore, the proposed method could reduce the computation time by 26%. Note that the CPU timeof post processing for channel estimation is negligible compared to the FDTD simulation. In this study, a PC with Intel Core i7 980X Extreme Edition and 24 GB memory was

OUCHIDA et al.: FAST MIMO CHANNEL CALCULATION TECHNIQUE FOR MULTI-ANTENNA SYSTEM USING SIGNAL MODULATION FOR FDTD METHOD

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used. A single-core compiler was used and the CPU time forthe model shown in Fig. 3 with 10000 time steps was 13,023
seconds.
4. Conclusion
This paper presented a fast FDTD analysis method for
multiple-antenna systems that uses signal modulation. The
modulation ensures orthogonality among the excited signals. Simulations showed that the proposed method yields
channel characteristics that agree well with those of the conventional method even though it reduces the computation
time by about 26%. Simulation of a small scale system,
2×2 antennas, demonstrated that our analysis method is effective in reducing the computation time of FDTD analysis
of multiple-antenna systems while oﬀering acceptable accuracy.
As a final remark, the proposed method can be applied to array antenna systems that have arrays larger than
2×2. Nevertheless, the increase of the transmitting antennas
causes the interference among the signals larger than that of
the 2 × 2 case. This means the quantitative studies on thesignal parameters, such as the length of the guard and symbol, and so on, are needed. This issue will be investigatedand presented in our future works.

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Acknowledgements
This research was partially supported by JSPS KAKENHI(25709030).
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Kazuma Ouchidareceived the B.E. inelectrical and electronic engineering from Iwate
University, Morioka, Japan in 2013. He is currently with East Japan Railway Company. He
received the APMC Best Student Paper Awardin 2013.

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 the APMC2014 Prize in 2014, respectively. His current research interest is MIMO antenna system and its applications. He is a member of IEEE.

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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.