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PAPER

Fast S-Parameter Calculation Technique for Multi-Antenna System
Using Temporal-Spectral Orthogonality for FDTD Method
Mitsuharu OBARA†a) , Student Member, Naoki HONMA† , Member, and Yuto SUZUKI† , Student Member

SUMMARY
This paper proposes an S-parameter analysis method thatuses simultaneous excitation for multi-antenna systems. In this method,
OFDM (Orthogonal Frequency Division Multiplexing) and CI (Carrier Interferometry) pulse generation schemes are employed for maintaining theorthogonality among the excited signals. In OFDM excitation schemes, thecharacteristics of the neighboring antennas can be calculated by assigningdiﬀerent frequency subcarriers exclusively. CI enables the simultaneousverification of the antennas distant enough since this method can provide
temporal orthogonality. Combining these two methods yields the simultaneous analyses of array antennas with both narrow and wide element
spacing. The simulation of a 2 × 2 multi-antenna shows that the resultsof the proposed method agree well with those of the conventional method
even though its computation speed is more 4 times that of the conventional
method.
key words: FDTD, OFDM, CI, S-parameter, multi-antenna

1.

Introduction

Recently, Multiple-Input Multiple-Output (MIMO) systems[1] and diversity antenna [2] are widely used in many areas of wireless communication. Diversity antennas are alsofrequently used to counter multipath fading. To design a diversity antenna with the desired characteristics, an eﬃcientsimulation method is needed. The FDTD (Finite-Diﬀerence
Time-Domain) method [3] is convenient for analyzing complicated structures including the antennas and propagation
environments. However, computation time rapidly becomes
excessive since it increases in proportion to the number ofantennas, since each antenna requires individual simulation.
For example, MIMO base stations have many antennas, andthe surrounding environments, such as the antenna tower
structure, and adjacent base stations, and so on, must be
taken into account. Therefore, a fast FDTD analysis for
multi-antenna systems is required.
In this paper, a fast FDTD analysis method that uses
simultaneous excitation is proposed for multi-antenna systems. In this method, OFDM (Orthogonal Frequency Division Multiplexing) pulse and CI (Carrier Interferometry) [4]are employed to maintain the spectrum and time orthogonality among the excited signals. The spectrum orthogonality
is used for calculating the characteristics of the neighboring
antennas by assigning diﬀerent OFDM subcarriers exclusively. CI enables the simultaneous verification of antennas
that are distant enough by controlling the power envelope ofManuscript received June 10, 2011.
Manuscript revised November 16, 2011.
†
The authors are with the Graduate School of Engineering,
Iwate University, Morioka-shi, 020-8551 Japan.
a) E-mail: t2310007@iwate-u.ac.jp
DOI: 10.1587/transcom.E95.B.1338

the pulse in the time domain. These methods can achievemuch faster analysis than the FDTD method with the conventional pulse. Section 2 details the concept of pulse generation and the analysis method. Section 3 shows the numerical analysis results, and clarifies the eﬀectiveness of the
proposed method is clarfied.
2.

Basic Concept

2.1 Orthogonality Using Frequencies
The OFDM system divides the available frequency band
into narrow subcarriers and multiplexes them orthogonally.
The frequency interval, Δ f , is given by
Δ f = 1/T

(1)where, T is signal length. If the frequency interval does
not meet this condition, the subcarrier interference is not insignificant. Figure 1 shows the conceptual sketch of the signal arrangement in the frequency domain. The antennas canbe analyzed by assigning signals to subcarriers alternately.
The signal numbers, 1 and 2 in Fig. 1 correspond to theantenna numbers, 1 and 2. The signal transformation is
given byN−1

S i (t) =k

{z(i−1)+nk exp( j2π f(i−1)+nk t)}

(2)

n=0

where N is the number of the subcarriers, n is the subcarrier index number, #i is the signal/antenna number, S i represents signals in the time domain, k is the number of antennas, z(i−1)+nk represents the signals in the frequency domain,

Fig. 1Signal arrangement.

c 2012 The Institute of Electronics, Information and Communication Engineers
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f(i−1)+nk is the center frequency of the subcarriers and {·}means the real part of the complex number. Additionally,transmitting the signals repeatedly can reduce the interference among the signals, since frequency orthogonality canbe maintained. In this paper, this method is used to obtainthe characteristics of the antennas that are placed close together.

must be raised. On the other hand, CI suits the situation
where the scattering parameter strongly depends on the frequency. For example, CI can be used for antennas that havevery narrow bandwidth. Therefore, either the OFDM or the
CI technique can be used depending on simulation modeland requirements.
2.3 Scattering Parameter Analysis

2.2 Temporal Orthogonality
Analysis using CI pulse can keep the orthogonality by exploiting the propagation time. In CI, the same amplitude
value is assigned to all subcarriers being analyzed. CI signals can be expressed aszl = e− j2π

Nt (l−1)N(3)where, zl is the signals in frequency domain, l is the index
number of subcarrier, N is the number of the subcarriers,
and Nt is linear phase oﬀset in time domain. The frequency
domain signals shown in (3) are transformed into the time
domain signals by (2). The CI method can collect most of
the signal powers in a short period of time.
Figure 2 shows the concept of analysis based on the CI
pulse. In this method, the antennas are separated by suﬃcient distance and excited together. The incident and arrived
signals can be separated by using the temporal orthogonality
thanks to the propagation time.
This paper uses this method to obtain the characteristics between the antennas that are suﬃciently separated. The
analysis that uses the CI technique exploits temporal orthogonality only to avoid the interference among pulses. Therefore, this method does not suﬀer from degradation in spectral resolution. When 2 × 2 antennas are simulated by the
OFDM technique, the spectral resolution becomes 1/2 thatof 1 × 1 antennas. In order to obtain same resolution with
only the OFDM technique, the time-step number in FDTD

This section describes a method to calculate the scatteringparameter in the FDTD method. In the FDTD method, thetotal voltage and current in the time domain can be calculated first. After that, the voltage and current waveform in
the time domain are transformed into frequency domain values by Fourier transformation. The scattering parameter isobtained from the voltage and current in the frequency domain. The complex envelopes of inward and outward propagating waves in the frequency domain are expressed as,1(u/Z0 1/2 + iZ0 1/2 )2
1
b = (u/Z0 1/2 − iZ0 1/2 ),
2

a=

(4)
(5)

Z0 is the reference impedance to define the scattering parameters, and u and i denote the voltage and current vector in thefrequency domain, respectively. The relationship among thescattering parameters, a, b is expressed asb = Sa(6)

where S is the scattering parameter matrix [5]. The scattering parameter is obtained as
S k j = bk /a j

(7)where, j and k are the port numbers of antennas. S k j is thek-th line and j-th column component of S. To calculate allcomponents of S, only one port out of all ports should be
excited in the conventional method. Therefore, individual
analyses for all ports are required.
3.

Numerical Results

3.1 Analysis Condition

Fig. 2 Sketch of the antennas and signals in the analysis that uses the CI
method: (a) conceptual sketch of the antennas, (b) observed time domain
signal.

This section analyzes the scattering parameters of 1×1 or
2×2 biconical antennas. Figure 3 shows the analysis models. Figure 3(a) is the model for the analyses using OFDMor CI technique, where antenna element spacing is D. The
dimensions of the analysis region are (D + 0.68) m × 0.44 m
× 0.305 m. Figure 3(b) shows the model for the combined
OFDM and CI technique, where the two arrays comprising
two close antennas are located at the ends of the analysis region. The dimensions of the analysis region are (D+0.68) m
× 0.78 m × 0.305 m. The length of the biconical antennas
is 0.105 m and two or four antennas are arranged symmetrically in the analysis region. N0 is defined as the number of time steps corresponding to the symbol length, andIEICE TRANS. COMMUN., VOL.E95–B, NO.4 APRIL 2012

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Fig. 3

Table 1Analysis model.

Parameters for analysis.

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

Biconical antenna
Delta-gap feed
5 mm
PML
4
0 ∼ 5 GHz
9.1477 ps

Fig. 4N0 = 1024 in this simulation. Therefore, the propagation distance during one symbol signaling is expressed as,
L = c(N0 Δt), where c is the wave velocity in a vacuum, Δt
is the time step. In this paper, the propagation distance during one symbol signaling L is 2.810 m. Details of the FDTD
analysis configurations are summarized in Table 1.
In the proposed method, the total number of subcarriers, N, is 1024, the frequency spacing, Δ f , is 0.106 GHz.
For comparison, the Gaussian pulse is used and two or fourindividual analyses are performed in total. N1 is the number of the necessary time steps; it depends on the separation
distance of the two facing arrays, i.e.,
N1 = (D + L)/cΔt.

(8)

In the following discussion, the ideal results obtained by
the Gaussian pulse excitation are called the conventional
method.
3.2 Analysis Using OFDM TechniqueFigure 4 shows OFDM pulse waveforms. Antenna 1 transmits signal 1 shown in Fig. 4(a), and antenna 2 transmits
signal 2 shown in Fig. 4(b). In the proposed method, analyses that use OFDM pulses require 2N0 time steps for excitation and analysis. This is because exciting an OFDM pulse
repeatedly can reduce the intersymbol interference.

OFDM pulse waveforms.

To evaluate the inter-subcarrier interference of the proposed method, simulations with one and two excitations arecarried out and compared. Similarly, the results of the conventional method that uses Gaussian pulse with N0 and 2N0time steps are compared.
Figure 5 shows the RMS (Root Mean Square) error of
the scattering parameters with one (N0 ) and two (2N0 ) excitations. Here, the symbol length is set to N0 Δt. The antenna spacing, D, is changed from 0.01L to 1.0L. The error is calculated by comparing to the result obtained from4096 time steps analysis with Gaussian pulse. The resultsin Fig. 5 show that the longer propagation distance between
the antennas becomes, the higher RMS errors of the scattering parameter become. This tendency is seen in both theconventional and proposed methods with N0 time steps.
There are two reasons that cause this error. One is
the propagation time between two antennas. When D/L ishigher than 0.7, the RMS errors in S 21 become greater than
30% when the conventional method with N0 time steps isused. Since the transmitted signal from one antenna cannot reach another antenna, the S 21 cannot be calculated correctly. Another reason is the orthogonality among the subcarriers, and this problem happens only in the proposed
method. The RMS errors of S 21 with N0 time steps are
greatly large compared with that with 2N0 time steps in the
proposed method. The length of the Fourier transform win-

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Fig. 6 Comparison of the simulated scattering parameters: conventional
method and proposed method using OFDM technique (D/L = 0.05).

Fig. 5 RMS error of scattering parameter versus antenna distance, D, in
OFDM technique.

dow is N0 time steps and the signal must be continuously
observed during the window span. Since the lack of the signal in the time domain degrades frequency orthogonality,
the RMS error is increased when the symbol is transmitted
just once.
By contrast, the proposed method with 2N0 time steps
can calculate scattering parameter with the RMS errors lessthan about 10%. In this case, the window is positioned at
the last N0 time steps, since higher accuracy is expected
when the voltage and current with steadier state are used.
It is found that this technique can improve the RMS error in
S 21 by more than 30% compared with the proposed methodwith N0 time steps. According to these results, the proposed
method can calculate the scattering parameter of two antennas with half computation time of the conventional method
in total.
Figure 6 shows the frequency characteristics of thescattering parameter at D = 0.05L. The scattering parameters are successfully obtained by the proposed method andagree very well with those of the conventional method inthis simulation.
3.3 Analysis Using CI TechniqueFigure 7 shows the CI pulse waveform. The antennas 1 and2 transmit the same signal shown in Fig. 7. The linear oﬀset
Nt is set to 21. As indicated by the Fig. 7, most of powers of
CI pulse are collected in the short period of excitation pulse.
After this, array spacing D is changed from 0.1L to 2.0L
by 0.1L step. Figure 8 shows the RMS error of the scatter-

Fig. 7

CI pulse waveform.

Fig. 8 RMS error between conventional method and proposed method
using CI technique.

ing parameters of the proposed method compared with thatof the conventional method. The results of Fig. 8 show thatS 21 has large RMS errors if D/L is lower than 1.5. This isbecause array spacing D is not large enough to maintain orthogonality in the time domain. By comparing the results in
Fig. 5, it can be seen that the OFDM scheme with 2N0 timesteps yields lower RMS error that the CI scheme. However,the spectral resolution of the OFDM scheme is half that of
the CI scheme. This simulation assumed biconical antennas,
which have broad bandwidth. That is, the scattering parameter variation in the frequency is not significant, and high

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Fig. 9 Comparison of simulated scattering parameters between conventional method and proposed method using CI technique (D/L = 2.0).

Fig. 11 RMS error between conventional method and proposed method
using combined OFDM and CI technique.

3.4 Analysis Using Both OFDM and CI TechniqueFig. 10

Waveforms using combined OFDM and CI technique.

spectral resolution is not required. When the antennas havenarrow bandwidth, such as patch antennas and inverted-F
antennas, the CI scheme would be used to evaluate the frequency characteristics of these antennas.
Figure 9 shows the scattering parameters using CI
pulses with D/L = 2.0. The scattering parameters obtainedby the proposed method agree very well with that of the conventional method.

In this scheme, the OFDM technique is used for the simultaneous analysis of the two neighboring antennas, and the
CI technique is applied to the two distant arrays. However,CI technique can not be applied to the OFDM pulse without modification if the subcarriers are assigned alternately
to two neighboring antennas. Therefore, the CI technique isapplied to alternate subcarriers in this simulation. Figure 10
shows the waveforms of combined OFDM and CI method.
Since a CI pulse is combined with an OFDM pulse the signal
exhibits two peaks in each symbol period. In this simulation,antennas 1, 3 are excited by signal  1 shown in Fig. 10(a).
Similarly, antennas 2, 4 are excited by signal  2 shown in
Fig. 10(b). Figure 11 shows the RMS error of the scatteringparameters of the proposed method using combined OFDM
and CI technique compared with the conventional method.
There are large RMS errors in S 31 and S 41 if D/L is lower
than 1.5. This is because array spacing D is not enough tomaintain orthogonality in the time domain.
In contrast, the results of S 21 have errors of around 10
to 20%. This is caused by the degradation in frequency orthogonality. When the CI technique is used, the number of
excitation symbols must be at least one since the CI tech-

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firm that the proposed analysis methods can greatly reducethe computation time of multi-antenna simulations.
4.

Conclusion

A fast FDTD analysis method for multi-antenna systems
was proposed in this paper. This method exploits orthogonalities in both the frequency and time domains. Simulations showed that the proposed method outputs scatteringparameters that agree well with those of the conventional
method even though it is four times faster. The results shownin this paper prove that our proposed analysis method is effective in reducing the computation time of FDTD analysis
of multi-antenna systems.
As a final remark, the proposed method can be applied
to array antenna systems that have more than 2 × 2 antennas
by using frequency orthogonality. But this will degrade thefrequency resolution. This problem needs further investigation.
Acknowledgements
This research is partially supported by MEXT KAKENHI(22760261).
References

Fig. 12 Comparison of simulated scattering parameters yielded by conventional method and proposed method using combined OFDM and CI
technique (D/L = 2.0).

Table 2

Method comparison.

nique exploits the temporal orthogonality and requires fastattenuation of the signal. As shown in Fig. 5, the RMS error
increases if only one symbol is excited. Even when the CI
technique is combined with the OFDM technique, a better
result can be obtained because in CI technique concentrates
the power of the signal into the beginning and middle of thesymbol. Figure 12 shows the scattering parameters yield by
the combined OFDM and CI technique; D is set to 2.0L.
The scattering parameters obtained by the proposed method
agree very well with those of the conventional method. Additionally, the proposed method can reduce total computation time by 75% compared to the conventional method.
Finally, Table 2 shows a comparison of the conventional method and our proposed methods. These results con-[1] G.J. Foschini and M.J. Gans, “Capacity when using diversity at transmit and receive sites and the Rayleigh-faded matrix channel is unknown at the transmitter,” Proc. WINLAB Workshop on Wireless Information Network, March 1996.
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[3] K.S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol.AP-14, no.4, pp.302–307, May 1966.
[4] K. Yokomakura, S. Sampei, H. Harada, and N. Morinaga, “A carrierinterferometry based channel estimation technique for one-cell reuse
MIMO-OFDM/TDMA cellular systems,” Proc. IEEE VTC 2006Spring, vol.4, pp.1733–1737, May 2006.
[5] J.W. Wallace and M.A. Jansen, “The capacity of MIMO wireless systems with mutual coupling,” Proc. IEEE 56th Veh. Technol. Conf.,
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Mitsuharu Obara
received B.E. degrees inFaculty of Electrical and Electronics Engineering from Iwate University in 2010. He is currently studying for M.E. degree at Iwate University. His research interests include FDTD analysis of multi-antenna systems.

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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, and theBest Paper Award of IEICE Communication Society in 2006, respectively. His current research interest is planar antennas
for high-speed wireless communication systems. He is a member of IEEE.

Yuto Suzuki
received B.E. degree in Facultyof Electrical and Electronics Engineering from
Iwate University in 2011. He is currently studying for M.E.degree at Iwate University. His research interests include beam switched antenna.