The Problem of the Fading Model Selection

IEICE TRANS. COMMUN., VOL.E84–B, NO.3 MARCH 2001

660

PAPER

The Problem of the Fading Model Selection Marcelo Agust´ın TANEDA† , Student Member, Jun-ichi TAKADA† , and Kiyomichi ARAKI† , Regular Members

SUMMARY Many experimentally and theoretically based models have been proposed to predict, quantitatively evaluate, and combat the fading phenomenon in mobile communication systems. However, to the best of the authors’ knowledge, up to now there is no objective method to determine which is the most suitable distribution to model the fading phenomenon based on experimental data. In this work, the Minimum Description Length (MDL) criterion for model selection is proposed for that purpose. Furthermore, the MDL analysis is performed for some of the most widely used fading models based on measurements taken in a sub-urban environment. key words: fading models, propagation model, Rayleigh, Rice, Nakagami, Weibull, fading model selection, MDL principle

1.

Introduction

In a radio communication system it is well known that when the signal is transmitted it degrades due to several factors; for example, the level of the signal decreases with the distance, the signal is further attenuated by objects placed in the medium, it may be affected by other signals or by noise, etc. In addition, the signal at the receiver antenna fluctuates due to various causes such as an object obstructing the radio path when the mobile moves into a specific area, multi-path propagation produced by reflection, refraction or diffraction of the electromagnetic waves on objects placed in the medium resulting in waves that combine constructively in some places and destructively in others, etc. As a result, the mobile moves through zones with different signal levels, causing in this way a fluctuation of the signal received. This fluctuation phenomenon at the receiver can be so severe as to produce a signal which is below the sensitivity of the receiver, thus rendering the communication impossible. The phenomenon is known as signal fading or just fading. Of course, the same effect takes place when the mobile is transmitting and the base station is receiving. Nowadays, the communication systems engineer has at his fingertips many models to simulate or predict the behavior of radio communication systems in the presence of fading. However, the question that arises is unavoidable: Which model should be used? Up to the present, many researchers have used the Rayleigh Manuscript received April 26, 2000. Manuscript revised September 20, 2000. † The authors are with Tokyo Institute of Technology, Tokyo, 152-8552 Japan.

distribution assuming that there is no worse case than that. This assumption has been proved to be wrong since there are cases where the Rayleigh predictions are too ideal [7]. In this paper, we are going to discuss the selection of the fading model based on experimental data by proposing the Minimum Description Length criterion to select the most suitable fading model. The selection will be based on data taken in the city of Sapporo, Japan, a sub-urban environment, and the models to be evaluated are the Rayleigh Model, the Rice Model, the Nakagami q Model, the Nakagami m Model, the Weibull Model, and what we call the generalized Nakagami Model. In what follows we will briefly present the MDL principle, the problem of parameter estimation and description length calculation for each model to be evaluated, the set up used for the experiment, and the results of the MDL selection method applied to the problem of the fading models. 2.

The Problem of the Model Selection

The problem of the model selection for the fading model can be explained by the use of Fig. 1. First of all, the kind of phenomenon we are going to study will be made clear by specifying the characteristics of the system, i.e. carrier frequency, analysis scope—macro if shading is considered or micro if it is not—, and the environment: rural, semi-urban or urban. In Fig. 1 there are several candidate models to be

Fig. 1

Fading model selection process.

TANEDA et al.: THE PROBLEM OF THE FADING MODEL SELECTION

661

evaluated. The model selection algorithm should compare each of the models with the actual phenomenon and evaluate them with the criterion of which model approaches the closest to it. Therefore, the choice of the selection criterion becomes an important issue. In our case, the Minimum Description Length criterion has been adopted because it is regarded as the most accurate for model selection cases [5]. Other criteria could be the Minimum Square Error criterion (MSE), the log-likelihood criterion or the information theory criterion proposed, among others, by Akaike (AIC) [5], [13]. 2.1 The MDL Principle The information theory criterion proposed by Schwartz and Rissanen consists of the calculation of an amount known as the Minimum Description Length (MDL) for each of the candidate models [5], [11], [12]. The model having the lower MDL value is selected as the best one among those evaluated. Each model will have different values of Description Length (DL) depending on the value of the parameters and on the specific data set composing the sample. To calculate the MDL, a parameter estimation process minimizing the DL for each model and each set of data should be performed. The description length is defined as: DL = − log P (R|Θ) +

1 k log N, 2

(1)

where R = [R1 , R2 , . . . RN ] is the sample’s vector, Θ = [θ1 , θ2 , . . . , θk ] represents the parameters of the model, P is the joint probability density function (PDF) of the model, also known as likelihood, k is the number of parameters and N is the number of elements in vector R (number of snapshots in the sample). The logarithm function can be calculated on any base as long as it is the same for all the models. For the sake of simplicity, the natural logarithm has been adopted in this work. The first term in Eq. (1) is called log-likelihood, and expresses the agreement between the model under study and the samples quantitatively. The second term in Eq. (1) represents the model complexity based on the model’s number of parameters [5], [11], [12]. 2.1.1 Parameter Estimation In Eq. (1) if the samples Ri are statistically independent, the joint PDF can be expressed as: P (R|Θ) =

N

P (Ri |Θ) ,

(2)

i=1

where P (Ri |Θ) represents the PDF of the model evaluated for the sample Ri . Since what we are looking for is the lowest MDL

value among the models to be compared, it seems only ˆ which mininatural to look for the parameters Θ = Θ mize the DL for each model. By using these estimated parameters in Eq. (1), the MDL for that model can be calculated. The best model can be selected by performing this procedure on all models, by comparing their MDL values and selecting the one having the lowest MDL. Thus, the interference of a subjective judgment is avoided [5]. 2.2 The Rayleigh Model The Rayleigh distribution is by far the most used one to model the fading phenomenon due to its simplicity, straightforward derivation from geometrical assumptions, and fairly good agreement with experimental data. The PDF is given by [1]–[4]: P (R|γ) =

R2 R − 2γ 2 e , γ2

(3)

where R is the level (amplitude) of the received signal and γ is the parameter of the Rayleigh distribution. After evaluating the PDF of the model for samples Ri , the following equation was obtained by performing the MDL parameter estimation: N 1 R2 . (4) γˆ = 2N i=1 i Finally, the MDL value can be obtained as: N 1 ln Ri M DL = N 1 + ln γˆ 2 − N i=1 +

1 ln (N ) . 2

(5)

2.3 The Rice Model As in the previous case, the Rice distribution—also known as the Nakagami n distribution—can be derived from geometrical assumptions, provides good agreement with experimental data for many cases, and is widely used. It differs from the former because it considers that in addition to all the Rayleigh assumptions there is a strong arriving wave—usually Line-Of-Sight (LOS) situations. Because of that, the model is not as simple as the previous one. The Rice PDF is given by [1]–[4], [9]: SR R R2 +S2 . (6) P (R|S, δ) = 2 e− 2δ2 I0 δ δ2 The estimated parameters Sˆ and δˆ can be calculated by solving the following two equations simultaneously:

ˆ Ri S N I 1 2 ˆ 1 δ , Sˆ = Ri

(7) N i=1 I Ri Sˆ 0

δˆ2


662 N 1 2 Sˆ2 + 2δˆ2 = R = R2 , N i=1 i

(8)

where I0 (∗) and I1 (∗) designates the Modified Bessel function of degrees 0th and 1st respectively. To solve Eq. (7) we used the Taylor series expansion of the Bessel functions’ division, which produced satisfactory results. After the estimated parameters have been calculated, the MDL value can be obtained by: N Sˆ2 1 2 ˆ − ln Ri M DL = N ln δ + N i=1 2δˆ2 N N Ri Sˆ 1 21 + Ri ln I0 N 2N δˆ2 δˆ2 i=1

i=1

+ ln N.

(9)

2.4 The Nakagami q Model This distribution is not frequently used in practice because it only applies in the case where there is not a strong wave arriving at the receiver, and when at the same time the in-phase and quadrature components of the received signal have different variances or are correlated. This situation is usually, but not always, limited to the case of the Not Line-Of-Sight (NLOS) propagation. Thus, the communications engineer usually prefers a model which can be applied to this case and to the case where a strong arriving wave is present as well such as the Nakagami m model as we will see later. The PDF of this distribution is given by [3]: 2R ( ) P (R|α, β) = √ e αβ 2 R 1 1 × I0 + . 2 α β 2 − R2

1 1 α+β

(10)

The estimated parameters can be calculated by solving the following system: 2

Ri 1 1 N I − 1 ˆ 2 α ˆ 1 β Ri2 , (11) α ˆ = R2 + N i=1 I Ri2 1 − 1 0

2

βˆ

α ˆ

2

Ri 1 1 N I − 1 ˆ 2 α ˆ 1 2 β Ri2 βˆ = R2 − R 1 1 N i=1 I0 i − 2 α ˆ βˆ = 2R2 − α. ˆ Finally, the MDL value can be calculated by: 1 ˆ ln α ˆβ M DL = −N ln 2 − 2 1 1 + + R2 α ˆ βˆ

(12)

+

2 N 1 Ri 1 − ln Ri + ln I0 2 βˆ α ˆ i=1

+ ln N.

(13)

2.5 The Nakagami m Model This model was proposed by Nakagami as an approximation to a more general equation derived from the assumption that the received signal follows a joint Gaussian distribution. In this work, we refer to such a distribution as the generalized Nakagami distribution (See Sect. 2.7). As explained before, the advantage of the Nakagami m model is that it can model cases where there is a strong arriving wave as well as cases where there is not a strong arriving wave. The Nakagami m model PDF is given by [3], [4]: 2 2 m m 2m−1 − mR R e Ω , (14) P (R|m, Ω) = Γ (m) Ω where Γ(∗) represents the gamma function. The estimated parameters can be obtained by solving the following system: N ˆ= 1 R2 , Ω N i=1 i

ˆ+ ln m ˆ − Ψ (m) ˆ − ln Ω

(15) N 2 ln Ri = 0, N i=1

(16)

where Ψ(∗) is the psi or digamma function. The MDL value can be obtained as: m ˆ Γ (m) ˆ −m ˆ ln −1 M DL = N ln ˆ 2 Ω N 2m ˆ −1 − ln Ri + ln N. (17) N i=1 2.6 The Weibull Model The Weibull distribution was introduced by Walodi Weibull at The Royal Institute of Technology of Sweden in 1939. He used it to estimate machinery lifetime. The Weibull distribution has also been applied in radar systems to model the dispersion of the received signal’s level produced by some types of clutter and recently it has been applied to model the fading phenomenon [7], [14], [15]. As in Appendix, Araki demonstrated that the Weibull distribution can also be regarded as an approximation to the generalized Nakagami distribution of the same order as the Nakagami m distribution. The Weibull model PDF is given by [7], [14]–[16]: c−1 R c c R e−( b ) . (18) P (R|b, c) = b b


663 Table 1 Mobile station side. Antenna Gain 5 dBi Cable loss 2.1 dB Antenna Type 5/8 λ + 3/4 λ monopole antenna Carrier frequency 1.5 GHz Band Mobile speed 40 km/h approx.

The estimated parameters can be obtained by solving the following system: N cˆ ˆbcˆ = i=1 Ri , (19) N −1 N N cˆ R ln R ln R i i i i=1 cˆ = − i=1 . (20) N cˆ N i=1 Ri The MDL value can be obtained by: N (ˆ c − 1) M DL = N cˆ ln ˆb − ln cˆ − ln Ri N i=1 N 1 cˆ + Ri + ln N. ˆbcˆN i=1

Table 2 Antenna Height Antenna Gain Antenna Type Sampling Rate

Table 3

Base station side. 30 m 15 dB 90◦ sector beam dual polarization patch antenna 1 milliseconds

Signal properties.

Transmitted Power Modulation

(21)

5W None

2.7 The Generalized Nakagami Model The generalized Nakagami model was obtained by Nakagami by assuming that the received signal is made up of two Gaussian components, the in-phase and the quadrature components. Therefore, the received signal follows a joint Gaussian distribution. The PDF of this distribution is given by [3], [4]:

π

−

R2 +A2

e σ2 −M cos θ 2 0 σ − M cos θ 2RA × I0 dθ. σ 2 − M cos θ

2R P (R|A, M, σ) = π

(22)

Since the PDF is given in integral form, it is not easy to obtain equations to calculate the estimated parameters, and therefore, a more complex optimization method should be employed. In this case we used the optimization toolbox of Matlab with satisfactory results [18]. The MDL value can be calculated by using the following equation: N ln Ri 2 + i=1 M DL = −N ln π N  2 +A2 i π − σ2R−M N cos θ e 1  + ln 2 N i=1 0 σ − M cos θ × I0

2Ri A σ 2 − M cos θ

dθ

+

3 ln N. 2 (23)

The set of models herein presented is the most widely used to model the fading phenomenon for the micro case. Other models such as the log-normal distribution or the Suzuki model—more suitable for macro cases—can be evaluated in a similar manner and are left for future studies.

3.

The Experiment

Measurements were taken in a sub-urban environment, where a car transmitted a signal in the band of 1.5 GHz. The car traveled at an approximate speed of 40 km/h in an area ranging from 200 m to 4 km from the base station. The system was set up according to the characteristics shown in Tables 1, 2 and 3. The experiment was performed with several antenna polarization configurations at the transmitter side as well as at the receiver side. However, the analysis showed that the propagation model is independent of the polarization characteristics. It should be clear for the reader that the methodology introduced in the previous sections is valid for different environments and analysis targets as shown in Fig. 1. Results from this and the following sections are only valid for sub-urban environment and microanalysis, where shading is not considered. Further experiments are required for other cases. 3.1 Data Acquisition As previously stated, the experiment consisted of a mobile traveling through a city, to be more precise, the city of Sapporo in Japan. This mobile transmitted a signal without modulation which was detected at the base station. The measurements were carried out at the base station when the car passed through 22 different locations in the city. While the car was traveling through each location 5000 measurements, spaced 1 milliseconds one from another, were taken. The experiment was repeated several times with different antenna configurations. The data measured was the level of the signal received corresponding to incoherent detection.


664

4.

Results

The samples obtained from each location were divided into sub-samples of 200 snapshots each, equivalent to a period of 200 milliseconds, 2.2 meters or 11 λ’s. The total number of sub-samples amounted to 12000. By using the procedures explained in Sect. 2 and after extensive parameter estimation, the MDL calculation process was performed for each sub-sample and for each model, thus giving as a result that each sub-sample has an associated MDL and parameter values for each model. The sub-samples were classified in terms of the range of the Weibull distribution parameter c associated to it with the following meaning: • 0 − 1.5: No strong wave arriving (usually NLOS), in-phase and quadrature components with different variances. • 1.5 − 2: No strong wave arriving (usually NLOS), in-phase and quadrature components with similar variances. • 2: It is the equivalent to the Rayleigh model case. • 2 − 2.5: One arriving wave is slightly stronger than the average (usually LOS with weak direct wave). • 2.5−: One arriving wave is much stronger than the average (usually LOS with strong direct wave). The results of the MDL calculations are shown; they indicate how many times each model was selected as the best for having the lowest MDL value. In addition to this, a table showing which model was the best when considering only the log-likelihood function (disregarding the number of model variables) is presented in Sect. 4.1. 4.1 Agreement Results The results of the log-likelihood analysis are shown in Table 4. In this table it should be noticed that since the Rayleigh, the Rice and the Nakagami q models are particular cases of the generalized Nakagami model, when one of such models is selected as the best for having the larger log-likelihood, the value of the log-likelihood for

Table 4

Best model according to the log-likelihood function.

Weibull parameter c Range Rayleigh Rice Nakagami q Generalized Nakagami Nakagami m Weibull

Total 0 − 1.5

1.5 − 2

2 − 2.5

2.5−

0 0

0 230

25 4083

10 1214

% 0.29 46.06

11

1142

190

10

11.28

10

1370

4231

1188

56.66

39 32

2432 268

1215 461

418 220

34.20 8.18

the generalized Nakagami model becomes equal. Therefore, it is also selected simultaneously as the best (both distribution counts increase by one on Table 4). The fact that there are no cases in which the generalized Nakagami model is selected as the best and others are not selected at the same time either indicates that these cases do not occur in practice. In other words: there are no cases in which the parameters A and M are simultaneously distinct from zero in Eq. (22). Therefore, the generalized Nakagami model is not expected to be selected as the best according to the MDL principle since the number of parameters for this model is three, against two or one for the other models. The alert reader will notice that the addition of the times in which Rayleigh, Rice and Nakagami q models are selected as best in Table 4 is slightly different from the times on which the generalized Nakagami model is selected as the best. The difference is due to the fact that it was not possible to calculate MDL values for the 100% of samples for the generalized Nakagami model due to the complex numerical calculations. The number of times in which it was not possible to calculate MDL values for the generalized Nakagami model was lower than 1%. However, a study of some of those cases revealed that the MDL value obtained in the other distributions was also a local minimum for the generalized Nakagami distribution. It is also clear from this table that there are many cases in which the Nakagami m model provides better agreement than that provided by the generalized Nakagami model. This indicates that, despite the fact that the Nakagami m model was obtained as an approximation to the generalized Nakagami model, the two models do not cover the same space of cases. Another conclusion we can draw from this table is that the Rayleigh model, having only one parameter, was seldom selected as the best, as expected. The Weibull model does not provide good agreement either, being selected as the best in less than 9% of the cases. 4.2 MDL Results The results of the complete MDL analysis are shown in Table 5. For this case, the number of simultaneTable 5 Weibull parameter c Range Rayleigh Rice Nakagami q Generalized Nakagami Nakagami m Weibull

Best model according to the MDL principle. Total 0 − 1.5

1.5 − 2

2 − 2.5

2.5−

1 0

2479 4

3062 2188

10 1214

% 46.27 28.38

10

389

25

10

3.62

0

31

116

39

1.55

39 32

1003 195

521 175

418 220

16.51 5.18


665

ous model selection decreased noticeably. All the cases producing simultaneous model selection occurred when the generalized Nakagami model and a two-parameter model were selected. In this case, the generalized Nakagami model has a better agreement than the twoparameter model, but since the former requires three parameters, their MDL values are comparable. The fact that there are no simultaneous model selection cases between the two-parameter models indicates that they are mutually exclusive. Clearly, the Rayleigh model is selected as the best 46% of the times and the combination of the Rayleigh, the Rice, and the Nakagami m models covers more than 90% of the cases. The combination of the generalized Nakagami, the Weibull and the Nakagami q models scarcely covers 10% of the cases. The difference between the latter and the agreement results is that the MDL principle evaluates not only the agreement between the data and the models but also whether that difference justifies the use of a more complex model based on the number of each model’s parameters. 5.

Conclusions

In this work we have addressed the problem of the fading model selection by proposing the MDL principle to select the most suitable model based on experimental data. The study focuses on the fading phenomenon from the point of view of the statistical distribution of the signals received disregarding the time delay produced due to the various path lengths of the different waves received. From the results we can conclude that, up to the present, there is not one unique model which can provide all the characteristics of the fading phenomenon. The communications engineer should keep the complexity of the actual propagation characteristics in mind when designing or analyzing radio systems. An analysis based solely on the Rayleigh model will provide ideal results that occur less than 50% of the time. It can be concluded that the Rayleigh model will not always predict the worse case when calculating the bit error rate. It is clear from the results that there are many cases in which the BER is worse than that predicted by Rayleigh (when the Weibull parameter c is lower than 2) [7]. To obtain realistic results, it may be necessary to use one model combining several models or to propose a new model that behaves more like the actual phenomenon. The important issue in this work is the methodology used to evaluate different models quantitatively with respect to the actual radio wave propagation characteristics. A propagation model that behaves as the actual phenomenon does will enable us to accurately predict, simulate and design better radio communication systems.

Acknowledgements The authors would like to thank the Anten Co. and the Mitsui Bussan Co. for providing the data for the analysis and for granting permission to publish this paper. References [1] J.G. Proakis, Digital Communications, pp.716–728, McGraw Hill, New York, 1989. [2] Y. Okumura and M. Shinji, Basic Mobile Communications, pp.61–77, 161-186, IEICE, Tokyo, 1986. [3] M. Nakagami, S. Wada, and S. Fujimura, “Some considerations on random phase problems from the standpoint of fading,” The Institute of Electrical and Communication Engineers Proceedings, vol.36, no.11, pp.595–602, Nov. 1953. [4] M. Nakagami and S. Fujimura, “A proposal of the classification and the general representation of the field-intensity variations,” The Institute of Electrical and Communication Engineers Proceedings, vol.36, no.5, pp.234–240, Nov. 1953. [5] M. Wax and T. Kailath, “Detection of signals by information theoretic criteria,” IEEE Trans. Acoust., Speech & Signal Process., vol.ASSP-33, no.2, pp.387–392, April 1985. [6] M.A. Taneda, J. Takada, and K. Araki, “Comparison of performance between V-H polarization diversity, slanted polarization diversity and space diversity,” Proc. Int. Symp. Personal, Indoor, Mobile Radio Commun., pp.1388–1391, Osaka, Japan, Sept. 1999. [7] M.A. Taneda, J. Takada, and K. Araki, “A new approach to fading: Weibull model,” Proc. Int. Symp. Personal, Indoor, Mobile Radio Commun., pp.711–715, Osaka, Japan, Sept. 1999. [8] H. Suzuki, “A statistical model for urban radio propagation,” IEEE Trans. Commun., vol.COM-25, no.7, pp.673– 680, July 1977. [9] S.O. Rice, “Statistical properties of a sine wave plus random noise,” Bell Systems Technology J., vol.27, no.1, pp.109– 157, Jan. 1948. [10] S.O. Rice, “Probability distributions for noise plus several sine waves—The problem of computation,” IEEE Trans. Commun., vol.22, no.6, pp.851–853, June 1974. [11] J. Rissanen, “Modeling by shortest data description,” Automatica, vol.14, no.5, pp.465–471, 1978. [12] G. Schwartz, “Estimating the dimension of a model,” Ann. Stat., vol.6, no.2, pp.461–464, 1978. [13] H. Akaike, “Information theory and an extension of the maximum likelihood principle,” Proc. 2nd Int. Symp. Inform. Theory, suppl. Problems of Control and Inform. Theory, pp.267–281, 1973. [14] Special issue on mobile radio propagation, IEEE Trans. Veh. Technol., vol.37, no.1, Feb. 1988. [15] M. Sekine and Y.H. Mao, Weibull Radar Clutter, IEE, 1990. [16] M. Sekine, Radar Signal Processing Techniques, IEICE, pp.62–73, 1991. [17] H. Hashemi, M. McGuire, T. Vlasschaert, and D. Tholl, “A study of temporal variations of the indoor radio propagation channel,” IEEE Trans. Veh. Technol., vol.42, no.1, pp.733– 737, Aug. 1994. [18] T. Coleman, M.A. Branch, and A. Grace, Optimization Toolbox User’s Guide For Use with MATLAB, The MathWorks, 1999.


666

Appendix:

Comparison of Distributions by their Moments

Nakagami showed that the joint PDF of two real random Gaussian variables is related to its characteristic function by the Hankel transform as [3]: ∞ J0 (Rλ) p (R) dR, (A· 1) F (λ) = 0

where F (λ) represents the characteristic function, J0 (∗) is the Bessel function of the first kind and order 0 and p(R) is the joint PDF. By expanding the Bessel function in series form, Eq. (A· 1) can be written as: 2k ∞ (−1)k λ R2k , (A· 2) F (λ) = k!k! 2 k=0

where R2k represents the 2k-th moment of the distribution. From Eq. (A· 2) it becomes clear that the characteristic function and therefore the PDF are determined by the even moments of the distribution. Therefore, two distributions are equal if they have the same even moments. It is not difficult to find that the characteristic function of the Weibull distribution can be expressed by: 2k ∞ 2k (−1)k bλ Γ +1 , (A· 3) F (λ) = k!k! 2 c k=0

where Γ(∗) represents the gamma function. The moments of the Weibull distribution are given by:

n +1 , (A· 4) Rn = bn Γ c From these equations it becomes clear that by adjusting the parameters b and c it is possible to exactly approximate the Weibull distribution to other distributions up to the fourth moment—only the even moments are important since we are analyzing distributions in the [0, ∞) range. The analysis of the Nakagami m distribution shows that it can also approximate exactly other distributions up to the fourth moment; therefore, it has the same degree of approximation to the generalized Nakagami distribution as the Weibull distribution.

Marcelo Agust´ın Taneda was born in 1970 in C´ ordoba, Argentina. He received the Information Systems Engineer Diploma in 1995 and the Electronics Engineer Diploma in 1996 from the National Technological University of C´ ordoba, Argentina. At present he is finishing his M.S. degree in Electrical Electronics Engineering at the Tokyo Institute of Technology, Japan. His research interests are in the field of mobile communications including propagation, channel modeling and modulation techniques such as CDMA.

Jun-ichi Takada was born in Tokyo in 1964. He received the B.E., the M.E. and the D.E. degrees from the Tokyo Institute of Technology, in 1987, 1989 and 1992, respectively. From 1992 to 1994 he was a Research Associate at the Chiba University, Japan. Since 1994 he is an Associate Professor at the Tokyo Institute of Technology. He received the Excellent Paper Award and Young Engineer Award from IEICE Japan in 1993 and 1994, respectively. Dr. Takada is a member of ITEJ, IEEE, SIAM, AGU and ACES. His research interests are array signal processing, mobile communication and numerical simulation of waves.

Kiyomichi Araki was born in 1949. He received the B.E. degree in Electrical Engineering from Saitama University, in 1971, and the M.E. and D.E. degrees in Physical Electronics, both from the Tokyo Institute of Technology, in 1973 and 1978, respectively. From 1973 to 1975, and from 1978 to 1985, he was a Research Associate at the Tokyo Institute of Technology, and in the period 1985–1995 he was an Associate Professor at Saitama University. In 1979–1980 and 1993–1994 he was a visiting research scholar at the University of Texas, Austin and at the University of Illinois, Urbana, respectively. Since 1995 he has been a Professor at the Tokyo Institute of Technology. Dr. Araki is a member of IEEE and the Information Processing Society of Japan. His research interests are information security, coding theory, communication theory, circuit theory, electromagnetic field analysis and simulation, microwave circuit analysis and design etc.