New Analytical Models and Probability Density Functions for Fading in Wireless Communications

Transcription

1 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 6, JUNE New Analytical Models and Probability Density Functions for Fading in Wireless Communications Gregory D. Durgin, Member, IEEE, Theodore S. Rappaport, Fellow, IEEE, and David A. de Wolf, Life Fellow, IEEE Abstract This paper presents new envelope probability density functions (pdfs) that describe small-scale, local area fading experienced by narrow-band wireless receivers. The paper also develops novel pdfs that describe the local area fading of two specular multipath components in the presence of other diffusely propagating waves. These pdfs are studied in the context of classical fading pdfs such as the Rayleigh, Rician, and other distributions. Index Terms Envelope detection, fading channels, mobile communications, multipath channels, propagation, scattering. I. INTRODUCTION SMALL-SCALE fading experienced by a wireless receiver causes dramatic fluctuations in received signal strength as a receiver moves over a relatively small local area [1]. One of the most common methods for characterizing a fading channel is the use of a probability density function (pdf), which represents the probability density of the received signal strength. The shape of a pdf determines the performance of a wireless receiver in the presence of noise and interference [2], [3]. Proper characterization of fading pdfs also impacts the design and use of diversity schemes, equalization methods, and error-correction coding used for a communications link [4]. The summation of constant-amplitude waves with independent random phases provides a useful mathematical description of narrow-band local area UHF and microwave propagation [5] [7]. The CW complex baseband voltage,, as seen by a receiver has the following form: The s are the amplitudes of multipath waves and the s are their corresponding phases. For local-area propagation, the phase variables,, are treated as statistically independent random phase variables, uniformly distributed over the interval [0, 2 ) [8]. This is an excellent representation since the distances traversed by propagating waves are orders-of-magnitude greater than the wavelengths of a microwave-frequency carrier, so that phases are virtually independent and unpredictable at any given point within a local area [3], [9]. Paper approved by R. A. Valenzuela, the Editor for Transmission Systems of the IEEE Communications Society. Manuscript received September 10, 1999; revised July 6, The authors are with the Mobile and Portable Radio Research Group, Bradley Department of Electrical and Computer Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA USA ( gdurgin@ vt.edu; wireless@mail.utexas.edu; dadewolf@vt.edu). Publisher Item Identifier S (02) (1) This paper extends present theory and formulates new pdfs for the received signal magnitude or envelope of the voltage representation in (1). The envelope pdf is crucial because it determines the range of received signal strengths. Since received power is proportional to the square of received envelope, the envelope pdf can determine the ultimate Shannon channel capacity of a fading wireless link. We also show the incredible variety of pdfs that exists for the model of (1). New pdfs in this paper describe the widest range of fading behavior exhibited by (1) [10]. Original contributions of this work include the following: The analysis provides a physical explanation for why many narrow-band fading measurements defy characterization by the Rayleigh or Rician pdfs without violating the assumptions of local area stationarity, a phenomenon first observed for mobile radio by Suzuki in 1977 [11]. We generate new fading distributions that may result in worse overall performance for receivers than a comparable Rayleigh fading situation. The formulas in this paper are valuable for estimating wireless modem performance either analytically or by computer simulation. This work should help place new envelope pdfs and well-known canonical fading distributions into the same family of distributions that arise from the first principles of radio wave propagation. II. CALCULATION OF ENVELOPE PDFS This section reviews the basic method for generating envelope pdfs for received voltage in the form of (1). A. Reduced Wave Grouping In free space, it is possible to decompose the propagating waves into a collection of plane-wave voltages at the terminals of an antenna [12]. In a local area, those voltages are primarily due to homogeneous plane waves, with the sum being written in the form of (1) [13]. From this point, we can group parts of the summation in (1) into three different categories. Specular Component: A specular component is a single term,, in the summation of (1), representing one arriving multipath wave. The phase of a specular component is random, but the envelope is constant. Nonspecular Component: A nonspecular component is a group of two or more terms in the summation of (1), representing more than one multipath waves arriving at the receiver /02$ IEEE

2 1006 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 6, JUNE 2002 Diffuse Component: A diffuse component is a nonspecular component with numerous individual waves, each carrying power that is negligible compared to the total average power of the diffuse component. In a local area, it is possible to write the channel as the sum of a finite number of specular waves and a nonspecular component where is the number of specular waves. The grouping in (2) is arbitrary. Any number of specular components may be lumped inside or outside the nonspecular channel component. The reduced wave grouping, however, refers to a specific combination of specular and nonspecular wave components, according to the following definition [13]: A reduced wave grouping is a model of local area propagation in the form of (2) with the smallest value of coherent waves chosen such that the remaining nonspecular component is diffuse. In other words, a reduced wave grouping represents a local-area channel as the sum of a few large specular components and a single nonspecular term of left-over components that, together, are diffuse: where the term is the diffuse voltage. The criterion for what components we consider to be diffuse is flexible, depending on how accurately we desire to model the channel. The in-phase (real) and quadrature (imaginary) portions of the diffuse voltage are the sums of numerous small, independently phased waves. Such waves obey the central limit theorem, producing an in-phase and quadrature component that are identical, independent Gaussian random variables. Equation (3) may be written in the following form: where and are independent, zero-mean Gaussian random variables, each identically distributed with variance. Equation (4) is the basis for the rest of the analysis. B. Envelope Characteristic Functions Since (4) is a sum of independent random variables, the envelope pdf may be found by using a characteristic function approach. A pdf and its characteristic function are transform pairs. The method for generating a pdf for describing the random complex voltage of (4) consists of multiplying the characteristic functions of the individual diffuse term and specular terms. This product is inverse-transformed into a pdf describing the complex voltages of (4). (2) (3) (4) Normally, characteristic functions are designed to be Fourier transform pairs. However, we desire to calculate the envelope statistics of (4). Therefore, the envelope pdf,, and its characteristic function,, are defined as Hankel transform pairs, as derived in Appendix A [14] The characteristic function of a single specular wave with magnitude,,is and the characteristic function of the diffuse component with mean-squared voltage,, is [15] [17]. Thus, the general form for the envelope pdf of (4) is given by which is valid for. Equation (7) allows for any combination of single, constant-amplitude voltage waves and a group of voltage waves that carry diffuse power. As the next section demonstrates, (7) has a number of closed-form solutions that result from standard definite integrals [18], [19]. C. Catalog of Canonical Fading PDFs This section presents the five closed-form analytical pdfs that may be derived from (7). Table I lists the voltage envelope pdfs and provides supplementary information such as expressions for the mean value of envelope and envelope squared, the latter being proportional to average local-area received power [9]. 1) Rayleigh PDF: The Rayleigh pdf assumes that all multipath power is diffuse and occurs from the integration of (7) under the condition and nonzero. This definite integral is a standard result and produces the following pdf [19, p. 738]: Unlike the purely specular wave pdfs, the Rayleigh pdf is nonzero over the entire range of. The Rayleigh pdf has been used extensively to describe narrow-band local area fading for mobile radio receivers [3], [20], [21]. 2) One-Wave PDF: A trivial case of fading is the one-wave pdf, in which only one constant-amplitude wave is present in a local area. Its characteristic function, however, is a building block for other canonical fading pdfs. Integrating (7) for and produces a result of 0 for all values of except, which is infinite [18, p. 485]. Thus, the pdf is represented by where is an impulse function. The one-wave pdf, therefore, results in no envelope fading. 3) Two-Wave PDF: The two-wave pdf represents the envelope fading caused by the interference of only two constant-amplitude waves in a local area, corresponding to and (5) (6) (7) (8) (9)

3 DURGIN et al.: NEW ANALYTICAL MODELS AND PDFs FOR FADING IN WIRELESS COMMUNICATIONS 1007 TABLE I SUMMARY OF CANONICAL WAVE FADING ENVELOPE PDFS in (7). Integration of (7) under these conditions is a well-understood result which produces the following pdf [19, p. 718]: and, a result formulated by Nicholson in 1920 [23] and given as follows: or (10) Equation (7) evaluates to zero for and, leading to the limits placed on in (10). Fig. 1 plots several examples of the two-wave pdf and cumulative density function (cdf) using a convenient parameter,, which we have defined to relate the relative magnitudes of and to one another [10]. The -parameter ranges between 0 and 1 and is defined by Peak Specular Power Average Specular Power (11) As shown in Fig. 1, when the magnitudes of two multipath waves are equal,. In the absence of a second component ( or ),. For dissimilar voltage values, the two-wave pdf exhibits two prominent spikes which mark the interval over which the pdf is nonzero. For the limiting case of ( ), the lower spike disappears and the pdf permits envelope values of zero which correspond to complete destructive cancellation. 4) Three-Wave PDF: Two- and three-wave models often are used to describe fading in microwave digital radio communications [22]. The three-wave pdf in pdfs is the solution of (7) for. (12) The function is an elliptic integral of the first kind. Note that in (12) is a function of. Here we have used the subscript, which is appended to Nicholson s notation to avoid any confusion between this parameter and the -parameter used to describe the two-wave pdf. The values and define the interval over which the integration of (7) and, subsequently, the pdf is nonzero. They are given by (13) The expressions in (13) have an appealing geometric interpretation: the three-wave pdf is 0 for all such that four line segments of lengths,,, and are incapable of forming a quadrilateral [23]. As one might expect, the behavior of the three-wave pdf is varied and complicated. Fig. 2 plots just a few examples of the pdf and corresponding cdf. A comparison of the three-wave cdfs of Fig. 2 and the two-wave cdfs of Fig. 1 provides insight into the difference between specular and diffuse power. Unlike

4 1008 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 6, JUNE 2002 Fig. 1. Evolution of a two-wave pdf and cdf with varying 1. the two-wave case, the different plots of the three-wave cdf are much more similar to one another. This similarity is due to the central limit theorem: the addition of another constant-amplitude wave to two-wave propagation makes the total multipath power more diffuse. Despite the complex shape of the cdf, the general cases of three-wave propagation begin to approach the cdf for purely diffuse power. Thus, even if it were possible to analytically calculate a four-wave pdf, its usefulness would be limited since the most general cases will appear to be even more Rayleigh-distributed than the three-wave pdfs. 5) Rician PDF: The Rician pdf describes the fading of diffuse power in the presence of a dominant, specular multipath component [20], [24]. The analytical expression for the Rician distribution results from the integration of (7) under the condition and nonzero. After applying a well-understood definite integral relationship [19, p. 739], the resulting pdf is where is a zeroth-order modified Bessel function. (14) Fig. 2. Several cdfs and pdfs for three-wave local area propagation. Fig. 3 shows several different kinds of Rician pdfs and cdfs. The plots are labeled using a Rician factor, which is the ratio of the power of the dominant multipath component to the power of the remaining, diffuse multipath: Specular Power Diffuse Power (15) In the literature, the parameter is often given as a decibel value, which is of the quantity in (15). Notice from Fig. 3 that db corresponds to the Rayleigh pdf and the complete disappearance of the specular power. III. A PDF FOR TWO-WAVE WITH DIFFUSE POWER (TWDP) FADING If (7) is evaluated with and nonzero, then the two-wave with diffuse power (TWDP) pdf results [29]. We will use parameters similar to the physical Rician -parameter of

5 DURGIN et al.: NEW ANALYTICAL MODELS AND PDFs FOR FADING IN WIRELESS COMMUNICATIONS 1009 TABLE II EXACT COEFFICIENTS FOR THE FIRST FIVE ORDERS OF THE APPROXIMATE TWDP FADING PDF where Fig. 3. Evolution of a Rician pdf and cdf as the dominant multipath component increases. (15) and the two-wave -parameter of (11) to classify the shape of the TWDP pdf as (16) There is no exact closed-form equation for TWDP fading, but this section presents a family of closed-form pdfs that closely approximate the behavior of the exact TWDP pdf. The expressions are also helpful for quick numerical calculation, lending themselves to intensive simulation of fading communication links. A. Approximate TWDP Representation Appendix B derives the following family of pdfs that approximate the TWDP pdf: (17) We refer to the value in the summation as the order of the approximate TWDP pdf. By increasing the order in (17), the approximate pdf becomes a more accurate representation of the true TWDP pdf. However, using the first few orders ( through 5) yields accurate representations over the most useful range of and parameters. Table II records the exact coefficients derived in Appendix B for the first five orders of (17). The product of the parameters and determines which order of (17) should be used when representing TWDP fading. As the product of these two parameters increases, a higher order approximation is needed to model the TWDP pdf accurately. As a general rule of thumb, the minimum order is Order (18) Equation (18) is based on a graphical comparison between the approximate analytical functions and the true, numerical solution of the TWDP pdf. The approximate pdf will only deviate from the exact TWDP pdf if the specular power is much larger than the diffuse power (large value) and if the amplitudes of the specular voltage components are relatively equal in magnitude ( approaches 1). Despite being an approximate result, the family of pdfs in (17) have a number of extraordinary characteristics that are independent of order,, and parameters, and. They are mathematically exact pdfs. They integrate to 1 over the range. They are accurate over their upper and lower tails. These regions are important for modeling noise-limited or interference-limited mobile communication systems [2]. They all exactly preserve the second moment of the true pdf. The second moment is the most important moment to preserve since it physically represents the average local area power [9].

6 1010 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 6, JUNE 2002 Fig. 4. TWDP pdf and cdf for K = 0 db. They can be entirely described with three physically intuitive parameters. The physical parameters,, and as defined in this paper have straightforward physical definitions. They exhibit the proper limiting behavior. All of the pdfs contain, as a special case of, the exact Rician pdf and, as a special case of, the exact Rayleigh pdf. Accurate analytical representation of these pdfs reveals interesting behavior in fading channels that goes unnoticed using Rician pdfs, which are capable of modeling the envelope fading of diffuse power in the presence of only one specular component. B. Graphical Analysis Figs. 4 7 plot a series of pdfs and cdfs for TWDP fading. As shown by Fig. 4, there is little difference between the Rician pdf and the TWDP pdf when is less than 3 db. The difference gradually becomes more pronounced as increases, particularly when the specular power is divided equally between the two discrete components ( ). The db graph of Fig. 7 illustrates these distortions most dramatically. In fact, as the product of parameters and becomes large, the graph of the pdf becomes bimodal, exhibiting Fig. 5. TWDP pdf and cdf for K =3dB. two maxima. Conventional wisdom in wireless modem engineering states that a Rayleigh-distributed envelope is the worst case of fading, with signal strengths spending a great deal of time at low levels. Some TWDP fading channels in Fig. 7, however, show a random received envelope that reaches low levels more often than a Rayleigh fading distribution of comparable average power. C. Rayleigh and Rician Approximations For the limiting parameter cases of Table III, the exact TWDP pdf contains the Rayleigh, Rician, one-wave, and twowave pdfs. This demonstrates the generality of the exact and approximate TWDP pdfs. It also shows the utility of the threewave pdf, since it is the only analytical expression in pdfs that is not a general case of the TWDP pdf. Since the Rician and Rayleigh pdfs are special cases of the TWDP pdf, it is useful to know the range of parameters over which TWDP fading may be approximated by these simpler distributions. An inspection of the graphs of Figs. 4 7 reveals the range of and over which a Rician pdf approximates

7 DURGIN et al.: NEW ANALYTICAL MODELS AND PDFs FOR FADING IN WIRELESS COMMUNICATIONS 1011 Fig. 6. TWDP pdf and cdf for K = 6 db. Fig. 7. TWDP pdf and cdf for K =10dB. a TWDP pdf. In general, the TWDP pdf resembles a Rician pdf in shape for. Under this condition, the smallest of the two specular components may be grouped with the diffuse power so that only one large specular component remains. After computing a Rician -factor for this new grouping, the resulting Rician pdf will approximately describe the envelope of the TWDP fading. TWDP fading may be further approximated by a Rayleigh pdf if, in addition to the above-mentioned criterion, the power of the largest specular component is less than the power of the smaller specular component plus the average diffuse power: (19) This condition derives from Fig. 3, which shows that Rician pdfs resemble the shapes of Rayleigh pdfs (after scaling) for a Rician -factor less than 0 db. Under this condition, the entire sum of voltage components in (4) may be treated as diffuse power, despite the presence of two specular components. TABLE III THE TWDP PDF CONTAINS THE RAYLEIGH, RICIAN, ONE-WAVE, AND TWO-WAVE PDFS AS SPECIAL CASES The Rician and Rayleigh approximation conditions, therefore, are best summarized in terms of the TWDP and parameters by the following: Rician Condition: (20) Rayleigh Condition: (21) These conditions show the parameter range over which a TWDP pdf may be approximated by an analytically simpler Rician or

8 1012 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 6, JUNE 2002 TABLE IV THREE EXAMPLES OF TWDP FADING THAT MAY SIMPLIFY TO RAYLEIGH OR RICIAN PDFS Rayleigh pdf. If these conditions are not met, then the only recourse is to use (17) or some other evaluation of (7) for and nonzero. Table IV shows three examples of TWDP fading and determines the simplest approximate pdf that describes the voltage envelope of each. Case A in Table IV satisfies the Rayleigh condition of (21). Case B, on the other hand, satisfies only the Rician condition of (20). Case C satisfies neither condition and may not be approximated by a Rayleigh or Rician pdf. Note how the changes in voltage amplitudes between the three cases drastically affects the overall shape and calculation of the pdf, emphasizing the need for careful and accurate representation of TWDP pdfs. D. TWDP PDF Applications The TWDP pdf and its approximations are important for characterizing fading in a variety of propagation scenarios. Smallscale fading is characterized by the TWDP pdf whenever the received signal contains two strong, specular multipath waves. While this may occur for typical narrow-band receiver operation, directional antennas and wide-band signals increase the likelihood of TWDP small-scale fading. The use of directional antennas or arrays at a receiver, for example, amplifies several of the strongest multipath waves that arrive in one particular direction while attenuating the remaining waves [25]. This effectively increases the ratio of specular to diffuse received power, turning a Rayleigh or Rician fading channel into a TWDP fading channel. Wide-band signal fading will likely exhibit TWDP fading characteristics for similar reasons. A wide-band receiver has the ability to reject multipath components that arrive with largely different propagation time delays [26]. This property of a wide-band receiver will naturally separate specular multipath components from other diffuse multipath waves. Under these circumstances, the ratio of specular to diffuse received power increases for a given propagation delay and a TWDP fading channel may result. E. Closing Remarks on TWDP Fading Beyond the TWDP pdf, a three-wave with diffuse power (3WDP) pdf is the next logical step. The value of such an analytically difficult pdf, however, is questionable. Much like the previously discussed four-wave pdf, the central-limit theorem would begin to dominate the behavior of (4), making it difficult to distinguish between the different cases of a 3WDP pdf. For example, a 3WDP pdf may be approximated by the TWDP pdf if the smallest of the three specular voltage components is grouped with the diffuse power. This approximation would fail only if the diffuse power were small compared to the third smallest specular component yet such a situation implies that the diffuse power is so small that it could be ignored: a 3WDP pdf could then be approximated by the three-wave pdf. Therefore, it is safe to say that the analytical expressions of (17) and pdfs provide a near-complete description of the possible envelope fading of complex voltages in the form of (4). IV. CONCLUSION This paper has presented characterizations of local area fading according to the model of (4). The cases of 0, 1, 2, and 3 specular multipath components in the presence of other diffuse waves were studied to produce new families of pdfs. A tremendous variety of fading behavior can be modeled without resorting to empirical distributions that ignore the physics of propagation and often have more analytical complexity than the closed-form expressions in this paper. Coupled with parameter estimation and hypothesis testing techniques, these pdfs may lead to new ways of analyzing fading measurements [2]. The expressions of pdfs and (17) provide a nearly complete set of analytical models for studying local area narrow-band fading. The approximate TWDP pdfs and subsequent analysis are the most important, original contributions in this paper. As Table III demonstrates, these pdfs capture the widest breadth of fading behavior and provide an analytical alternative for describing local area fading that does not conform to classical pdfs. APPENDIX A DERIVATION OF ENVELOPE CHARACTERISTIC FUNCTIONS In the study of pdfs, it is convenient to define a characteristic function, which is the Fourier transform of the pdf [27]. The standard mathematical definitions for finding a characteristic function,, from a pdf,, and vice versa are given below: (22) (23) Characteristic functions are useful for studying the addition of independent random variables. If random variables,, and

9 DURGIN et al.: NEW ANALYTICAL MODELS AND PDFs FOR FADING IN WIRELESS COMMUNICATIONS 1013 satisfy the relationship and and are independent, then their characteristic functions satisfy the relationship [28]. Characteristic functions are also useful for studying the superposition of two independent random voltages. Since voltage,, is complex-valued, its characteristic function must be a double Fourier transform over the joint pdf of the random in-phase,, and quadrature,, voltage components ( ). This transformation is demonstrated as follows: APPENDIX B DERIVATION FOR TWDP FADING PDFS Following from (7), the TWDP pdf results from the integration of (29) The following relationship is used to transform the integration [19, p. 758]: (24) (25) Starting with the basic envelope pdf,, as a function of only an envelope,, it is possible to extend this pdf into a joint pdf using the relationship (26) Equation (26) is a joint pdf, albeit in terms of envelope,, and phase,, variables instead of in-phase,, and quadrature,, variables. Equation (26) assumes that the net phase,, is uniformly distributed, independent of entirely consistent with the small-scale local-area propagation model. Rather than convert (26) into an joint pdf, it is more convenient to make a change of variables in the transform definition of (24). With the polar-coordinate substitutions,, and, (24) becomes (27) This result is substituted into (29) to produce (30) (31) Integration with respect to produces the following equation, in terms of the and parameters of (15) and (11) [19, p. 739]: (32) Although still not in closed form, (32) is more intuitive and insightful than (29), somewhat resembling the Rician pdf. This form is useful for generating alternate or approximate representations for the TWDP pdf. A. Approximate Representation The left side of (32) looks remarkably similar to the Rician pdf. The difficulty in evaluating this pdf lies in the integral term on the right side: Equation (27) may be regrouped: (33) (28) where and. The angle is unimportant since the integration of is over the entire period of the cosine function in (28). Thus, the dependence of the characteristic function is solely dependent on the variable. The bracketed term in (28) is a standard definite integral that evaluates to a zeroth-order Bessel function of the first kind [19]. The final expression for the transformation from envelope pdf to characteristic function is (5). Using a similar set of reductions, the reverse transformation from characteristic function to envelope pdf becomes (6). The only assumption made in these transformations is the statistical independence and uniform distribution of the complex voltage phase. To generate the th-order approximation to the TWDP pdf, the integration of (33) is performed by representing the integrand as a Legendre polynomial with terms. The Legendre polynomial representation for this integrand, assuming that the function is sampled at uniform increments, is Integrand of Equation (33) where (34)

10 1014 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 6, JUNE 2002 The integration of this result,, may be written as pdf behaves as approaches 0. The function collapses to a zeroth-order modified Bessel function (35) where (40) The approximation turns the complicated integration of (33) into simple polynomial integrations of. To simplify the representation of this approximate PDF, it is useful to define the following helping function,, which is defined in (17). Equipped with this relationship and the symmetrical property of ( ) it is now possible to write the approximated integration as When this result is substituted into (17), once again the Rician distribution results. Regardless of order, each approximated pdf contains the exact Rician pdf as a special case. D. Preservation of the Second Moment The second moment, E, of a fading envelope is essential to the calculation of mean power. It can be shown that (36) This leads to (17) and provides a simple method for approximating this complicated pdf to an arbitrary level of precision. B. Property as a PDF By making an approximation when constructing a PDF, it is often possible to destroy the mathematical properties of a PDF. Of principle concern, the PDF may no longer integrate to unity. It can be shown that (37) which is valid for any combination of and. Mathematically, the integrand in (37) appears as two half -Rician distributions, which explains their integration to 1. The integration of the total approximated pdf is then given by Furthermore, the coefficients have the following property: (38) (39) Regardless of order, each approximated pdf integrates exactly to 1 over the interval. C. Proper Limiting Behavior The exact distribution for TWDP fading collapses to a Rician function as approaches 0. Now notice how the approximated (41) which is valid for any combination of and. The integration of over the total approximated pdf is then given by E (42) In terms of voltages, the value becomes, which is the second moment of the true TWDP pdf. Regardless of order, each approximated pdf preserves the true second moment of the exact TWDP pdf. REFERENCES [1] R. H. Clarke, A statistical theory of mobile-radio reception, Bell Syst. Tech. J., vol. 47, pp , [2] A. J. Coulson, A. G. Williamson, and R. G. Vaughan, A statistical basis for lognormal shadowing effects in multipath fading channels, IEEE Trans. Commun., vol. 46, pp , Apr [3] J. D. Parsons, The Mobile Radio Propagation Channel. New York: Wiley, [4] J. G. Proakis, Digital Communications, 2nd ed. New York: McGraw- Hill, [5] J. F. Ossanna, A model for mobile radio fading due to building reflections: Theoretical and experimental fading waveform power spectra, Bell Syst. Tech. J., pp , Nov [6] W. C.-Y. Lee, Statistical analysis of the level crossings and duration of fades of the signal from an energy density mobile radio antenna, Bell Syst. Tech. J., pp , Feb [7] G. D. Durgin and T. S. Rappaport, A basic relationship between multipath angular spread and narrowband fading in a wireless channel, Electron. Lett., vol. 34, no. 25, pp , Dec. 10, [8] T. S. Rappaport, Characterization of UHF multipath radio channels in factory buildings, IEEE Trans. Antennas Propagat., vol. 37, pp , Aug [9], Wireless Communications: Principles and Practice. Englewood Cliffs, NJ: Prentice-Hall, 1996.

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Tech. J., vol. 27, no. 1, pp , Jan [25] J. H. Winters, Smart antennas for wireless systems, IEEE Personal Commun., vol. 1, pp , Feb [26] W. R. Braun and U. Dersch, A physical mobile radio channel model, IEEE Trans. Veh. Technol., vol. 40, pp , May [27] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. New York: McGraw-Hill, [28] H. Stark and J. W. Woods, Probability, Random Processes, and Estimation Theory for Engineers, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, [29] R. Esposito and L. R. Wilson, Statistical properties of two sine waves in Gaussian noise, IEEE Trans. Inform. Theory, vol. IT-19, pp , Mar Gregory D. Durgin (M 02) was born in Baltimore, MD, on October 23, He received the B.S.E.E., M.S.E.E., and Ph.D. from the Virginia Polytechnic Institute and State University (Virginia Tech), Blacksburg, in 1996, 1998, and 2000, respectively. He performed his graduate work at the Mobile & Portable Radio Research Group (MPRG) in radio wave propagation, channel measurement, and applied electromagnetics. He is currently a Japanese Society for the Promotion of Science (JSPS) Post-Doctoral Fellow at Morinaga Laboratory in Osaka University, Osaka, Japan, where he performs research in space time channel modeling. He has published 20 technical papers in international journals and conferences. He also serves regularly as a consultant to industry. Dr. Durgin received the 1998 Blackwell Award for best graduate research presentation in the electrical and computer engineering department at Virginia Tech. He received the 1998 Stephen O. Rice Prize, with co-authors T. S. Rappaport and H. Xu, for best original research paper published in the IEEE TRANSACTIONS ON COMMUNICATIONS. Theodore S. Rappaport (S 83 M 83 SM 90 F 98) received the B.S.E.E., M.S.E.E., and Ph.D. degrees from Purdue University, West Lafayette, IN, in 1982, 1984, and 1987, respectively. From 1988 to 2002, he was a member of the electrical and computer engineering faculty, Virginia Polytechnic Institute and State Universith (Viriginia Tech), Blacksburg, where he was the James S. Tucker Professor and founder of the Mobile & Portable Radio Research Group (MPRG), one of the world s first university research and teaching center dedicated to the wireless communications field. He recently joined the University of Texas (UT) as the William and Bettye Nowlin Chair in Electrical Engineering and is director of the newly formed Wireless Networking and Communications Group on UT s Austin campus. In 1989, he founded TSR Technologies, Inc., a cellular radio/pcs manufacturing firm that he sold in He formed Wireless Valley Communications, Inc., in 1995 and relocated the company to Austin in He has 28 patents issued or pending and has authored, co-authored, and co-edited 18 books in the wireless field, including the textbooks Wireless Communications: Principles & Practice (Englewood Cliffs, NJ: Prentice-Hall, 1996, 2002), Smart Antennas for Wireless Communications: IS-95 and Third Generation CDMA Applications (Englewood Cliffs, NJ: Prentice-Hall, 1999). He has co-authored more than 200 technical journal and conference papers. Since 1998, he has been series editor for the Prentice-Hall Communications Engineering and Emerging Technologies book series. He serves on the Editorial Board of International Journal of Wireless Information Networks and the Advisory Board of Wireless Communications and Mobile Computing. He also serves as chairman of Wireless Valley Communications, Inc., an in-building/campus design and management product company. He has consulted for over 25 multinational corporations and has served the International Telecommunications Union as a consultant for emerging nations. Dr. Rappaport received the Marconi Young Scientist Award in 1990, an NSF Presidential Faculty Fellowship in 1992, the Sarnoff Citation from the Radio Club of America in 2000, and the James R. Evans Avante Garde award from the IEEE Vehicular Technology Society in He is active in the IEEE Communications and Vehicular Technology Societies. He is a registered professional engineer in the state of Virginia and is a fellow and past member of the board of directors of the Radio Club of America. David A. de Wolf (SM 65 F 88 LF 00) received the B.Sc. degree in physics and mathematics and the Dutch Doctorandus degree in theoretical physics, both from the University of Amsterdam, the Netherlands, in 1955 and 1959, respectively, and the Doctorate degree in technology from TU Eindhoven, The Netherlands, in From 1962 to 1982, he was with RCA Laboratories, David Sarnoff Research Center, Princeton, NJ, where his earlier work covered various aspects of radio-, radar-wave, and optical scattering from atmospheric media and objects, and also wave propagation through irregular media. Subsequently, he worked on electron-optics applications to TV systems, specifically with respect to guiding electron beams accurately in complicated electromagnetic fields. In 1982 he joined the Electrical Engineering Department, Virginia Institute of Technology and State University (Virginia Tech), Blacksburg. Since then he has worked on problems concerning scattering in particulate media, especially regarding propagation of radar and optical waves through rain, fog, haze, and other aerosols. He has been associated in the summers with TU Delft s International Research Centre for Telecommunications-Transmission and Radar (and predecessor organizations) since He is the author of a text on elecctron optics and of an undergraduate text on electromagnetic theory. Dr. de Wolf is a fellow of the Optical Society of America and a member of the American Association of Physics Teachers, the Dutch Physical Society, Sigma Xi, and Eta Kappa Nu, as well as Commissions B and F of the USNC of the International Union of Radio Science.