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Methods for Modeling of Specified and Measured Multipath Power-Delay Profiles Matthias Pätzold, Senior Member, IEEE, Arkadius Szczepanski, and Neji Youssef, Member, IEEE
Abstract—In this paper, five fundamental methods are proposed to model the multipath power-delay profile of frequency-selective indoor and outdoor wireless channels. Three of them are new, and the other two are well known, but their performance, however, has not been studied in detail up until now. All procedures are universally valid so that they can be applied to any specified or measured multipath power-delay profile. The performance of the proposed methods is investigated with respect to important characteristic quantities such as the frequency correlation function (FCF), average delay, and delay spread. The method found to perform best is -norm method (LPNM). This procedure is applied the so-called to measurement data of multipath power-delay profiles collected in different propagation environments. It is shown that the realization complexity of tapped-delay line-based simulation models for fading channels can be reduced considerably by using the LPNM. This is a great advantage for the development and specification of channel models for future wireless systems. Index Terms—Deterministic channel modeling, mobile radio channels, multipath channels, multipath power-delay profile, tapped-delay line model.
I. INTRODUCTION
T
HE MOBILE radio channel is a combination of propagation paths, each with its own attenuation, phase distortion, and temporal dispersion, which manifests itself as frequency-selective fading. The impact of the frequency selectivity of the channel on the transmitted signal increases, the higher the transmitted data rates. In order to characterize temporal dispersive outdoor and indoor wireless channels, the so-called delay spread is commonly used. The delay spread is defined as the square root of the second central moment of the multipath power-delay profile. In time division multiple access (TDMA) systems, this quantity, for instance, plays a key role in determining the degree of intersymbol interference and, thus, determines whether an adaptive equalizer is required at the receiver or not. In code division multiple access (CDMA) systems, the delay spread also determines the number of resolvable paths and the effectiveness of the RAKE receiver. In other words, understanding the behavior of frequency-selective channels and temporal dispersion is of primary importance to the radio system designer and has a great impact on the performance of the communication system. Therefore, from Manuscript received December 7, 2000; revised November 30, 2001. M. Pätzold is with the Faculty of Engineering and Science, Agder University College, N-4876 Grimstad, Norway (e-mail:
[email protected]). A. Szczepanski is with Baseband DSP Software, Siemens AG, D-38228 Salzgitter, Germany (e-mail:
[email protected]). N. Youssef is with the Ecole Supérieure des Communications de Tunis, Ariana, 2083 El Ghazala, Tunisia (e-mail:
[email protected]). Digital Object Identifier 10.1109/TVT.2002.801747
the beginning of mobile communications, every effort has been made to find realistic models for the multipath power-delay profile and for the characteristic quantities derivable from it. Material on measurement results for multipath power-delay profiles and delay spreads can be found, e.g., in [1] and [2]. Theoretical investigations on this topic have been done recently, e.g., in [3] and [4]. Specifications of continuous multipath power-delay profiles for channel models developed for the global system for mobile communications (GSM) can be found in [5]. A suitable and often used model that enables the simulation of multipath fading channels is the discrete-time tapped-delay line model [6], [7]. Such a simulation model can be interpreted as a transversal filter of order with time-varying tap gains. The problem now is to determine the parameters of the simulation model in such a way that the multipath power-delay profile characteristics of the channel simulator are, for a given order , as close as possible to the desired, i.e., measured or specified, multipath power-delay profile characteristics. Finding a solution to this problem is exactly the topic of the present paper. Altogether, we will discuss five different parameter computation methods with different degrees of sophistication, where four of them are purely deterministic methods and one is a statistic procedure. The procedures are called the method of equal distances (MED), mean-square-error method (MSEM), method of equal areas (MEA), Monte-Carlo method (MCM), and -norm method (LPNM). For the first time, the MSEM, the MEA, and the LPNM are used to compute the tapped-delay line parameters. The MED and the MCM are well-known standard methods, but their performance has not been investigated in detail previously. In our paper, each method is formulated in universally applicable terms. Moreover, to demonstrate the strengths and weaknesses of the proposed methods, we employ each procedure to the exponential multipath power-delay profile as specified for the typical urban (TU) channel in [5]. Here, the performance of each procedure will be evaluated analytically with respect to the frequency correlation function (FCF), the average delay, and the delay spread. The numerical results of the paper show that the LPNM has the best performance and the MCM the worst. A duality relation, concerning the parameters of the Doppler power spectral density and the multipath power-delay profile, is noted for the considered simulation model. To achieve our objectives described above, we have organized the paper as follows. Section II reviews the characteristic quantities of multipath power-delay profiles with one-sided negative exponential shapes. This type of profile is considered as reference profile throughout the paper. Section III investigates the multipath power-delay characteristics of tapped-delay line channel models. Section IV is devoted to the derivation,
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analysis, and discussion of five different fitting procedures allowing the adaptation of the multipath power-delay profile of the tapped-delay line model to any given theoretical or measured multipath power-delay profile. Section V presents the results of the comparison between the proposed modeling procedures. Applications of the outperforming method to measured profiles are described in Section VI. Finally, the conclusions are drawn in Section VII. II. MULTIPATH POWER-DELAY PROFILE REFERENCE MODEL
OF THE
Bello’s [8] wide-sense stationary uncorrelated scattering (WSSUS) model has widely been accepted as an appropriate stochastic model for time-variant and frequency-selective wireless channels. This model, which is valid for most radio channels, assumes that the received scattered components with different propagation delays are statistically uncorrelated and that the correlation properties of the channel are stationary. The WSSUS model is completely characterized by the scattering , which is a two-dimensional function, denoted by function of the propagation delay and the Doppler frequency . From the scattering function one can derive the multipath by computing the integral of power-delay profile over the Doppler frequencies, i.e., (1)
Fig. 1. Tapped-delay line model for a frequency-selective mobile fading channel in the equivalent complex baseband.
As reference profile, we consider the TU channel model specified in [5]. For this channel model, the multipath power-delay is given by profile (5)
else
s and MHz. The real-valued constant has been introduced here to normalize the . The average power in the profile to unity, i.e., and the delay spread of the TU channel are delay s and s, respectively. given by Using the relation (2), the FCF corresponding to the TU channel is then obtained as where
The Fourier transform of the multipath power-delay profile is referred to as the FCF [8] (2) which is a measure of the frequency coherence of the channel. There are two important quantities that allow us to describe the characteristics of multipath power-delay profiles. One is the avand the other quantity is the delay spread . erage delay is the first moment of , i.e., The average delay (3)
(6) Finally, it should be mentioned that the TU model given by (5) and (6) can be modified to describe also the rural area (RA) channel model specified in [5] by merely changing the values , , and . of the constants III. MULTIPATH POWER-DELAY PROFILE SIMULATION MODEL
OF THE
The tapped-delay line channel simulator considered here is characterized by the following time-variant impulse response (7)
and the delay spread , i.e., moment of
is the square root of the second central where is the number of paths with different propagation delays, are discrete propagation delays, the real-valued coefficients are called the path gains, and (8) (4)
is an important quantity, beIn particular, the delay spread enables the characterization of the channel with recause gard to temporal dispersion or frequency selectivity. The recipis a measure of the correlation bandwidth; rocal value of that is, the frequency interval over which the frequency response of the channel does not change significantly.
is a zero-mean complex stochastic Gaussian process or, alternatively, a zero-mean complex deterministic Gaussian process [9]. The design of deterministic Gaussian processes with given spectral shapes (Doppler power spectra) was the topic of several papers, e.g., [10] and [11]. The equivalent complex baseband structure of the simulator for the frequency-selective channel model with the impulse response described by (7) is shown in is the input signal, is the resulting Fig. 1. In this figure, ( ) output signal, and
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stands for the delay difference between the propagation paths and . Since the value of the first propagation delay has no effect on the behavior of the simulation model, it has been set to zero for convenience. In [12] and [13], it has been shown that of the channel model defined by the scattering function (7) can be expressed as (9) denotes the Doppler power spectrum of the where . complex stochastic (or deterministic) Gaussian process Without loss of generality, we may assume that the variance of is unity, so that is normalized in such a way holds. In this case, the multipath that of the tapped-delay line channel simpower-delay profile ulator can be expressed by
on the FCF is contained in one half of the period of the FCF. for , the real and imaginary Finally, since are related to each other by the Hilbert transform parts of (14) , , and denote the real part, the imagwhere inary part, and the Hilbert transform, respectively. It should be mentioned that (13) and (14) also apply to the FCF of the reference model. Now, using (3) and (4) in conjunction with (10), we can exand the delay spread of the press the average delay simulation model as follows:
(15)
(16)
(10) is a finite sum of weighted delta functions, Hence, where the delta functions are located at and the corresponding weighting factors are given by the squares of the path gains ( ). Comparing the above equation with the resulting expression for the Doppler power spectral density derived in [11] for deterministic channel models, it turns out that the multipath power-delay profile and the Doppler power spectral density stand to each other in a duality relation. Taking the of the simuFourier transform of (10) results in the FCF lation model. Thus (11) are highIn the following, some properties of the FCF is periodic with the period given lighted. The function by (12) denotes the greatest common divisor.1 Thus, we where , where is an integer. Note can write as . Furthermore, it that follows from (2) that the FCF exhibits the Hermitian symmetry property, i.e., (13) From (12) and (13), it also follows that and , where and are integers. Thus, the real and imaginary parts of the FCF are even and odd functions, respectively, and the FCF is Hermitian symmetric with respect to one half of the period, i.e., . Consequently, the complete information to the value
gcdf~ ~ . . . ~ g is defined as = 0 1 . .. ; L 0 1 and gcdf ~ ~ . . . ~ g = 1 .
1Here, the greatest common divisor ; ; ; follows. Let q , where q are integers for ` ; ; is a real-valued constant, then ; ; ;
1
~ = 11
Considering the expressions (10) and (11) and (15) and (16), we realize that the multipath power-delay characteristics of the simulation model can be investigated analytically provided that and are known. the parameter sets IV. PARAMETER COMPUTATION METHODS It is the task of this section to compute the sets and in such a way that the multipath power-delay characteristics of the simulation model are close to those of the reference model. Altogether, five different methods are presented for the computation of the discrete propagation delays and the proppath gains . All methods are exemplified using agation paths, which enables a fair comparison with the performance of the 12-path COST 207 models. Nevertheless, it should is just exemplary. Later in Secbe stressed that the case tion V, the methods are all compared over a larger range of values. A. MED The MED was originally presented in [12]. Here, this procedure will be considered as a reference method when discussing the performance of other, more sophisticated procedures. Let be any given specified or measured multipath poweris equal to zero delay profile characterized by the fact that . Then, the discrete propagaoutside the interval tion delays are defined by multiples of a constant quantity according to (17) is related to and according to . From (12) and (17) it follows that the FCF is a periodic function in the frequency separation variable with . the period where
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Next, we partition the interval into subin, where , tervals according to for , and . Furthermore, we impose on our and with rechannel model that the integrals over are identical, i.e., spect to
(18) Thus, after substituting (10) in the right-hand side of (18), we obtain the following for the path gains: (19) Hence, the square of the path gain of the th discrete propagawithin tion path is a measure of the average delay power apthe subinterval . It is shown in Appendix A that if tends to infinity, i.e., for proaches . Fig. 2(a) shows the reference multipath power-delay profile according to (5) in comparison with the multipath powerof the simulation model by applying the delay profile . Substituting (5) in (2) and afterwards taking MED with leads to the results shown the absolute value of the FCF in Fig. 2(b). For comparison, we have also presented the corresponding analytical results of the simulation system in the same figure. These results have been obtained by substituting (17) and . More(19) in (11) and then taking the absolute value of over, Fig. 2(c) shows the convergence behavior of the average [see (15)] and the delay spread [see (16)] as a delay function of the number of discrete paths .
(a)
(b)
B. MSEM The basic idea of the MSEM is to determine the sets and in such a way that the mean-square error of the FCF (20) . The is minimal within a given frequency interval , will be upper limit of this interval, denoted here by defined below. A closed-form solution to this problem exists are given if, and only if, the discrete propagation delays is given by by (17). Hence, the set , where . As a consequence of the equally spaced discrete propagation delays, is a periodic function in with it follows that the FCF . One half of this period is, therefore, a the period proper value for the upper limit of the integral in (20). Thus, . we choose Inserting (11) into (20) and setting the partial derivatives of with respect to each path gain the mean-square error equal to zero, i.e., , results in the following closed-form expression for the path gains (21)
(c) Fig. 2. (a) Multipath power-delay profile S ( ) of the TU channel. (b) Absolute value of the FCF jr ( )j. (c) Average delay D as well as the delay spread D in comparison with the corresponding quantities of the simulation model when using the MED.
for all , where . By substituting (21) in (10), it can be shown that the multipath of the simulation model approaches power-delay profile
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the desired profile if the number of discrete paths tends if ( ). However, to infinity, i.e., , where for finite values of , we have to write , and ( ). As an example, we apply the MSEM to the TU multipath power-delay profile as defined by (5). The results obtained for of the simulation model the multipath power-delay profile are shown in Fig. 3(a) and the absolute value of the FCF and (b), respectively. Here, the number of discrete paths was . This enables a direct comparison with the equal to results shown in Fig. 2(a) and (b). Obviously, the quality of the over the interval is approximation slightly better for the MSEM than for the MED. The average and the delay spread are both shown in Fig. 3(c) delay as a function of the number of discrete paths . A closer investigation of the MSEM and the MED reveals that the equal spacing of the discrete propagation delays seriously restricts the performance and the efficiency of these methods.
Note that when the -path tapped-delay line model is implemented on a computer, the discrete propagation delays have to be quantized in such a way that the greatest common divisor is equal to the sampling interval of the resulting digital channel simulator. However, if the sampling interval is sufficiently small, then the period of the FCF—given —is in general much larger than the period obby tained by using the MED or the MSEM. The impact of the quantization of the discrete propagation delays on the performance of the simulation model is discussed in Section IV-E. Analogously to Figs. 2 and 3, the Fig. 4(a) and (b) show the results for the multipath power-delay profile and the FCF, respectively. The results presented for the simulation model with nonquantized discrete propagation delays were obtained by substituting (22) and (27) in the closed-form expressions (10) and (11). Particularly, as shown in Fig. 4(c), the average delay and the delay spread converge relatively slowly in comparison with the corresponding results obtained by applying the MED [see Fig. 2(c)] and the MSEM [see Fig. 3(c)].
C. MEA The MEA was originally derived in [11] intending to model the Doppler power spectral density. Due to the duality relation between the Doppler power spectral density of deterministic processes and the multipath power-delay profile of the tappeddelay line model, the MEA can also be used to determine the path gains and the discrete propagation delays. Using the MEA, we choose the same constant quantity (22) for all path gains and determine the discrete propagation delays in such a way that the area under the multipath powerover the interval equals for all delay profile , i.e.,
D. MCM The design of frequency-selective mobile fading channels using the Monte-Carlo principle was first introduced in [14], refined in [15], and improved in [16]. The basic idea of the MCM is to generate the discrete propagation delays according to a given probability density function describing the distribution of the propagation delays . It can is proportional to be shown without any difficulty that , i.e., the multipath power-delay profile (28) is a real-valued constant defined by . According to [15], it is convenient to establish a uniformly distributed random generator with the and to perform a mapping outcomes
where
(23) . To solve this problem, we first introduce an auxilwith , which is defined by iary function (24)
(29) in such a way that the cumulative distribution function of the discrete propagation delays is equal to the desired cumulative . As shown in [17], distribution function is the inverse of so that the the function discrete propagation delays are given by
Applying (23), we may then write
(30) (25)
The application of the MCM to the TU multipath power-delay profile [see (5)] results in the following formulae for the discrete propagation delays and path gains :
of exists, then a general expresIf the inverse function sion for the discrete propagation delays is given by
(31a)
(26)
(31b)
Now, we apply this procedure to our reference model by substituting (5) in (25). After solving the resulting equation for , we finally obtain (27)
for . Computing the sets and for and putting the obtained coefficients in (10) and (11) leads to the results shown in Fig. 5(a) and (b), respectively. Due to the fact that the elements of the set are outcomes of a random generator, it follows from that the characteristics of the mapping
where
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(a)
(a)
(b)
(b)
(c)
(c)
Fig. 3. (a) Multipath power-delay profile S ( ) of the TU channel. (b) Absolute value of the FCF jr ( )j. (c) Average delay D as well as the delay spread D in comparison with the corresponding quantities of the simulation model when using the MSEM.
Fig. 4. (a) Multipath power-delay profile S ( ) of the TU channel. (b) Absolute value of the FCF jr ( )j. (c) Average delay D as well as the delay spread D in comparison with the corresponding quantities of the simulation model when using the MEA.
and strongly depend on the actual realization of . However, the approximations the elements of the set and [see Fig. 5(c) for ] are
poor, even for large values of . Nevertheless, in the limit one can show that tends to and, thus, and . A further characteristic
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E. LPNM An appropriate error function for our purpose is the defined by
-norm
(32)
(a)
(b)
(c) Fig. 5. (a) Multipath power-delay profile S ( ) of the TU channel. (b) Absolute value of the FCF jr ( )j. (c) Average delay D as well as the delay spread D in comparison with the corresponding quantities of the simulation model when using the MCM.
property of the MCM is the fact that for , the stawith respect to the random variable tistical average of leads to the desired result, i.e., . Conseand quently, we can write .
and is where denotes a real-valued number a sufficiently large quantity limiting the upper bound of the inteby , so that gral. Here, we define the upper limits of the integrals appearing in (20) and (32) are identical. This is the prerequisite for a fair comparison with the performance of the MSEM. However, detailed investigations for different types of multipath power-delay profiles have revealed that, in general, sufficiently good results can be obtained if is larger than twice that value, i.e., . The task we are confronted with is to optimize the elements of the and , defining the shape of the FCF sets [see (11)], in such a way that the -norm (32) is minimal. A can be found numerically, e.g., by using local minimum of the Fletcher–Powell optimization algorithm [18]. This procedure requires proper initial values for the quantities and , which can be obtained, e.g., by using the MED or the MSEM. converges to in the limit In this case, it follows that and, thus, tends to . Consequently, we can for if . write It should be noted explicitly that the discrete propagation delays are included in the optimization procedure without imposing any boundary condition. In general, the optimization algorithm therefore results in real-valued quantities, which are not equally spaced. However, to enable the implementation of the tapped-delay line model on a computer, it is necessary to quantize the obtained optimized values , so that they have a greatest , which defines the sampling common divisor, denoted by interval of the digital channel simulator. Of course, the quantization of has an impact on the quality of the approximation and on the period . However, the is sufinfluence of the quantization effect can be ignored if . In this ficiently small, e.g., by choosing is , which is case, the period of more than 50 times the period obtained by using the MED and the MSEM by keeping the same number of discrete paths . In general, the LPNM results in a much higher period of the FCF than the MED and the MSEM. To demonstrate the power of the LPNM, we apply this procedure to the TU multipath power-delay profile [see (5)] by , , and . The obchoosing tained results are presented in Fig. 6(a)–(c). Especially by comshown in Fig. 6(b) with those presented paring the FCF in Figs. 2(b), 3(b), 4(b), and 5(b), we can realize that the LPNM has a much better performance than the four methods described in the previous subsections. Fig. 6(c) reveals that the approxi( ) is nearly perfect if . mation V. COMPARISON OF THE METHODS In this section, we compare the parameter computation methods (MED, MSEM, MEA, MCM, and LPNM), described
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(c) Fig. 6. (a) Multipath power-delay profile S ( ) of the TU channel. (b) Absolute value of the FCF jr ( )j. (c) Average delay D as well as the delay spread D in comparison with the corresponding quantities of the simulation model when using the LPNM with p = 2.
in the previous sections, by evaluating the mean-square error with respect to the FCF [see (20)], the average delay, and the delay spread as a function of the number of discrete paths
(c) Fig. 7. (a) Mean-square error of the FCF. (b) Average delay D . (c) Delay spread D of the TU channel in comparison with the corresponding quantities of the simulation model when using the MED, MSEM, MEA, MCM, LPNM, and the tapped-delay line model introduced by COST 207.
. Fig. 7(a) shows the mean-square error with respect to the FCF [see (20)] evaluated for the five methods with as a function of the number of discrete paths . Moreover, this figure shows the results obtained
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for both the original and the alternative 12-path COST 207 TU models with the model parameters (path gains and discrete propis also a function agation delays) given in [5]. Note that of . Fig. 7(b) and (c) illustrate the average delay and the delay spread, respectively. In these figures, the results for the 6-path and the 12-path COST 207 TU models also appear, with two points for each value—one for the original and one for the alternative COST 207 TU model. According to this comparison, it turns out that the LPNM is definitely the best method with respect to the three characteristic quantities. The reason why the LPNM works so well even at low numbers of paths is that this method exploits the full flexibility of the tapped-delay parameters. This is in contrast line model by optimizing to all other methods, where parameters are pre-defined and the remaining parameters have to be determined according to a method-specific criterion. Another result of the performance comparison is that the MCM appears to be worthless. It is a fact that the MCM is not “just another method”; it is a method that fails totally.
(a)
VI. APPLICATIONS In this section, we apply the LPNM to two different measured multipath power-delay profiles of equivalent satellite channels. The measurement campaign was initiated by the ESA and carried out by the Deutsches Zentrum für Luft- und Raumfahrt (DLR) in different propagation environments and under various elevation angles at a carrier frequency of 1.82 GHz [19]. Here, we use two measurements carried out in an open rural area and in an urban environment with elevation angles in the region of 10 –20 . The average delay and the ns and ns for delay spread are given by ns and ns the open rural area and by for the urban environment. Fig. 8(a) presents the multipath power-delay profile of the measured signal in comparison with the corresponding discrete profile of the simulation model for the open rural area. Fig. 8(b) shows the absolute value of the respective FCF corresponding to the measured channel and the simulation model. For the simulaand tion model, the LPNM has been used with , where is given by ns. In Fig. 9(a) and (b), the corresponding functions for the urban environment are presented. We have again applied the LPNM with and , whereas is in this ns. Note that in Figs. 8(b) and 9(b), case given by the FCF is represented up to one half of the period of the measured signal, which was equal to 80 MHz. The relative errors of the average delay and the delay spread are smaller than for both measurements. Thus, one can say that these characteristic quantities are approximately identical for the simulation model and the measured channel. Figs. 8(b) and 9(b) exhibit the fact that the FCFs of the simulation model approximate the corresponding functions of the measured channel very closely over . the interval Here, we have chosen a tapped-delay line model of order . According to , this results MHz [Fig. 8(b)] and MHz in
(b) Fig. 8. (a) Multipath power-delay profile S ( ). (b) Absolute value of the FCF jr ( )j of a measured channel in an open rural area in comparison with the corresponding functions of the simulation model when using the LPNM with L = 20.
[Fig. 9(b)]. For system studies or practical applications, howshould be adapted to the (total) bandwidth of ever, the transmitted signal of the system being simulated, e.g., by . In this case, the required number of taps defining is given by (33) denotes the smallest integer greater than or equal to where . Invoking some of the major cellular standards and assuming that the multipath power-delay profile is given by (5), then the for GSM ( kHz); required number of taps is for IS-95 ( MHz); and for W-CDMA MHz). ( VII. CONCLUSION The multipath power-delay profile of a tapped-delay line channel model of order is completely defined by two different
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a meaningful tradeoff between obtaining good performance at a given number of paths and ease of computing the model parameters. APPENDIX A. Derivation of the Relation Using the MED
for
In this appendix, we show that by using the MED, the multiof the simulation model tends to path power-delay profile of the reference model the multipath power-delay profile if the number of discrete paths tends to infinity. Substituting (19) in (10) and using the subinterval and the quantity , we obtain the following result in the limit (a)
(34) REFERENCES
(b) Fig. 9. (a) Multipath power-delay profile S ( ). (b) Absolute value of the FCF jr ( )j of a measured channel in an urban area in comparison with the corresponding functions of the simulation model when using the LPNM with L = 20.
kinds of parameters: the discrete propagation delays and the path gains. In this paper, we have described five fundamental methods (MED, MSEM, MEA, MCM, LPNM), which enable us to compute these parameters in such a way that the multipath power-delay characteristics of the designed channel model are close to those of a given (specified or measured) frequency-selective mobile fading channel. The MED, MSEM, MEA, and LPNM are entirely deterministic methods, whereas the MCM is basically a statistic procedure. The performance of the presented methods has been investigated analytically with respect to the frequency correlation function, the average delay, and the delay spread. It turned out that each of the deterministic procedures outperforms the MCM. The method with the best performance is definitely the LPNM. Concerning the LPNM, it should also be mentioned that the numerical computation expenditure of this top-performance procedure is higher than that of the other four methods. However, this disadvantage is of minor importance when using today’s high-speed computers. Concerning this point of view, engineers need not agonize over
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Matthias Pätzold (M’94–SM’98) was born in Engelsbach, Germany, in 1958. He received the Dipl.-Ing. and Dr.-Ing. degrees in electrical engineering from Ruhr-University Bochum, Bochum, Germany, in 1985 and 1989, respectively, and the Dr.-Ing. habil. degree in communications engineering from the Technical University of Hamburg-Harburg, Hamburg, Germany, in 1998. From 1990 to 1992, he was with ANT Nachrichtentechnik GmbH, Backnang, where he was engaged in digital satellite communications. From 1992 to 2001, he was with the Digital Communication Systems Department, Technical University Hamburg-Harburg. Since 2001, he has been a Professor of mobile communications with Agder University College, Norway. He is the author of Mobile Radio Channels—Modeling, Analysis, and Simulation (in German) (Wiesbaden, Germany: Vieweg, 1999) and Mobile Fading Channels (Chichester, U.K.: Wiley, 2002). His current research interests include mobile radio communications, especially multipath fading channel modeling, multi-input–multi-output (MIMO) systems, channel parameter estimation, and coded-modulation techniques for fading channels.
Arkadius Szczepanski received the Dipl.-Ing. degree in electrical engineering from Technical University of Hamburg-Harburg, Germany, in 1999. Currently, he is a System Engineer in the UMTS mobile phones research and development department of the Siemens AG, Germany.
Neji Youssef (M’95) was born in Tunisia. He received the B.E. degree in telecommunications from the “Ecole Superieure des Communications de Tunis,” Tunisia, in 1983, the “D.E.A.” degree in electrical engineering from the “Ecole Nationale d’Ingenieurs de Tunis,” Tunisia, in 1986, and the M.E., and Ph.D. degrees in communications engineering from the University of Electro-Communications, Tokyo, Japan, in 1991 and 1994, respectively. From 1994 to 1996, he was a Research Associate at the University of ElectroCommunications, Tokyo, Japan. Since 1997, he has been Assistant Professor at the “Ecole Superieure des Communications de Tunis,” Tunisia. His research interests include noise theory and statistical modeling of multipath fading channels.
