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A Complex Probability Density Function: An Application to Multipath Fading Channels in Wireless Communication Systems Chirasil Chayawan, Ph.D. Department of Electronics and Telecommunication Engineering King Mongkut’s University of Technology Thonburi, Bangkok, 10140 Thailand Phone: (662) 470-9063, Fax: (662) 470-9070 Email: [email protected] ABSTRACT An argument of common probability density function (pdf) is generally a real number, which describes the density, or mass in case of discrete random variable, of a random variable. However, in some applications, the pdf can be a complex function, so called complex probability density function. This kind of pdf is then, as general complex function, characterized by the amplitude and phase and is usually extensively used in quantum physics. In this paper the properties of complex pdf are investigated. The application to the multipath fading channels in wireless communication systems is also studied. Some models of multipath fading channels are given for supporting the validity of complex pdf in wireless communication systems. Keywords: Complex Probability Density Function, Superposition, Multipath Fading Channel 1. INTRODUCTION The probability density function (pdf) in general is a real function. This kind of pdf is widely used in many applications including the modeling of multipath fading channels in wireless communication systems. When the attribute of a pdf is real, the statistical interpretation is not complicated and the statistical property is then characterized in term of the amplitude or absolute value of that pdf. However, the notation of complex probability is not common in the literature, the concept and its considerations have been discussed in physics pertinent to the physical characteristics of tunneling probability amplitude functions [1], complex diffusion wave function [2], wave-matter interactions [3], etc. For the best of our knowledge, the application of complex pdf in multipath fading channels has been limited and need to be investigated in details. In this paper, therefore, the applications of complex pdf in multipath fading channels of wireless communication systems are examined. Some examples of valid conditions in wireless communications are given to support the usefulness of complex pdf, which turn out to be the general form of commonly used models such as Rice or Rayleigh distributed multipath fading channels. The organization of this paper is as follow. The concepts of complex pdf are investigated in section II. In section III, the application of complex in multipath fading channels is given. The analysis and conclusion are then provided in section IV and V, respectively. 2. COMPLEX FUNCTION PROBABILIT DENSITY The complex pdf in specific aspects does not signify as a pdf of a complex random variable, which is, in general, defined as a joint pdf of real and imaginary parts of corresponding complex random variable. In addition, the complex pdf is described as a probability density of any random variables; however, that density function takes a possibility to be complex. At this point, some confusion might be raised, since the word “density” or mass, in general, is a real measurable quantity. When the statistical property of a random process is characterized by a complex pdf, two questions are needed to answer properly [2]. i) What is the physical inference of complex pdf? ii) Does the complex pdf provide any measurable quantity? In general, the pdf of a random variable is, by definition, a properly normalized function that assigns a probability "density" to each possible outcome within some interval. For example, a pdf of a Gaussian random variable A with mean m and variance σ2 can be written as a − m 2 ) 1 − ( f A (a ) = exp ∈ R ; − ∞ ≤ a ≤ ∞, 2πσ 2 2σ 2 under the pdf curve cannot be calculated as a volume (unit of cube), due to the density function being a function of one variable a (comparing with the pdf of a complex random variable). In addition, the volume in figure 1 also includes a complex probability, which is impractical for probability value taking from zero to one in real axis. Therefore, in the case of a random variable having a complex density, the cdf can be evaluated by find the area under the real part of the corresponding pdf to satisfy the definition and relationship of cdf and pdf [5]. Im[fA(a)] Re[fA(a)] (1) which is always a positive real number and (1) is the one (many)-to-one transformation from a real plane to a real plane. In contrast, if f A (a ) is complex, i.e. f A ( a ) ∈ C; -∞ ≤ a ≤∞ , (2) The transformation is now an one (many)-toone mapping from a real plane to a complex plane, but all statistical properties are needed to maintain in the complex plane as that of in a real plane. Note that the relationship of a density function and a cumulative distribution function (cdf) should be primarily studied. An cdf of a random variable A in [(2)] is defined as the probability of a random variable that is less than or equal to a specific value of A, and can be obtained by integrating the density function of A, i.e., a FA (a ) = P ( A ≤ a ) = ∫ f A (α ) d α . (3) −∞ Also note that the cdf is nothing but is the area under the pdf curve that less than or equal to an upper limit a of the integration in (3). Returning to the complex pdf, the corresponding cdf should also be, by definition, the area under the complex pdf curve. Since the density is complex having both real and imaginary quantities, the area under the pdf curve does not satisfied the definition. Moreover, for the case of a complex pdf as shown in figure 2, the area a Fig.1: An example of a complex pdf of a random variable a, showing the real and imaginary parts of the complex pdf. However, what the significance property is that the imaginary value of the complex pdf contains. Note that the complex pdf of a real random variable especially in wireless communication systems is generated by a superposition of finite random variables and usually involves the movement, scattering, diffusion or diffraction [6], [7]. It can also be shown by [5] that the imaginary part is proportional to the degree of the correlation. The imaginary part is then a function of a correlation coefficient or other parameters that state the degree of the relationship of each individual random variable of the superposition of the random variable having a complex pdf. At this point, one can see that the envelope of the complex pdf might be insignificant to interpret since the real and imaginary part have diverse properties, i.e. one for pdf and the other for elementary correlation, respectively. Finally, the phase of the complex pdf needs to be revealed. It can be shown by Barkay and Moiseyev [2] that the phase of the complex probability amplitudes plays an important role in the interference among resonance states during scattering experiments. In other words, it associated with the phase of the resonance channel probability amplitudes. 3. AN APPLICATION OF COMPLEX PDF IN MULTIPATH FADING CHANNELS In this section, the application of complex pdf will be studied for multipath fading environment in wireless communication systems. The complex pdf will be used to analyze the exacted envelope pdf for multipath fading channels. First the theory of multipath fading is summarized. Then the individual phase of each channel, which lies beneath the probability assumptions of the complex random signals, is investigated and shows that for some specified phase distributions resulting the complex envelope pdf, which turns out to be more general than a classical assumption of uniform distribution. In wireless communication systems, a multipath fading channel can be modeled as a superposition of sinusoidal signals. The inphase and quadrature-phase components of the resultant vector are given by [6] N X c + jX s = ∑R n (cos ϕn + j sin ϕn ) (4) n =1 where there are N random vectors (representing the number of scatterers in the channel), each with corresponding amplitude Rn and phase ϕn. When the amplitude Rn and phase ϕn of each individual vector are independent, and it is known that the pdf of phase ϕn is either i) uniformly distributed in any interval of 2π or, ii) nonuniformly distributed but its pdf is symmetrical about zero, the resultant envelope and phase, which can be given by R= X c + X s θ = tan 2 2 −1 Xs Xc (5) are independent, and also the quadrature components X c and X s are uncorrelated regardless of the distribution of Rn [1], [7]. When the number of random vectors is sufficiently large ( N → ∞) , the quadrature components are Gaussian distributed based on the central limit theorem (CLT). Then since Xc and Xs are jointly uncorrelated, they are also independent and the resultant envelope is Rayleigh distributed. There are many pdf’s that are usually applied as the individual phase pdf, fϕn (ϕn ) such as uniform, Simpson, Cardiod and von Mises distribution [9]. The significant phase distribution in this literature is the von Mises distribution that is commonly used in modeling angle distributions in man applications. This distribution is generally used in an application of directional statistics, and first studied for measuring the atomic weights from integral value [10]. It has also been widely used as a noise model in oscillators [11], [12], and the angle of arrival in antenna array systems [10]. The pdf of phase ϕn when it is distributed by the von Mises distribution is given by [9] fϕ (ϕ ) = 1 exp κ cos (α (ϕ − µ)) , 2π I 0 (κ ) ( ) µ − π ≤ ϕ ≤ µ + π, (6) Where the parameter µ is the mean direction, the parameter κ is known as the concentration parameter, I n ( x ) is an n-th order modified Bessel function of the second kind [13], and α is the number of modes of the von Mises distribution, which, in this case, represents the number of concentrated direction of signals at the receiver. Figure 2 compares the von Mises with Cardiod phase distribution with uniform distribution as a special case. It can be seen that Cardiod distribution can be approximated, for a small value of κ, by von Mises pdf. The Cardiod distribution then may be obtained by [9] κ C µ , ~ M ( µ , κ ) , 2 (7) where C (*,*) and M (*,*) stand for Cardiod and von Mises, respectively. Therefore, it is reasonable to use the von Mises distribution to model the elementary phase of the fading channel in the remainder of this paper. When the von Mises distribution is used to modeled the individual phase of multipath fading channel in wireless communication systems, the exact envelope distribution can be evaluated by [1] f R (r ) = r ×e ∞ N ∫ ∫0 ∏ n=1 jrn q cos(ϕn −ϑ ) ∞ 0 f Rn (rn ) ∫ 2π 0 d ϕn drn J 0 (qr )qdq , fϕn (ϕn ) (8) 0.4 Von Mises Cardiod --------- 0.35 κ=1 4. 0.3 0.5 fϕ(ϕ ) 0.25 0.2 0.2 0.15 0 0.5 0.1 0.05 0 -3 -2 -1 0 1 2 3 ϕ Fig.2:: Density of the von Mises distribution π for µ = and κ = 0,1, 3, 8,12 . 2 ϑ = tan−1 (ωc , ω s ) where is the angle between Xc and Xs in frequency domain, f Rn (rn ) is the pdf of the elementary envelope and J 0 ( ⋅) is the Bessel function of the first kind of order zero. Substituting (6) in (7), and evaluating the inner integral, yields J= 2π ∫ 0 ∫ 2π × 0 = pϕn (ϕn ) e jrn q cos(ϕn −ϑ) d ϕn = 1 2π I 0 (κ ) exp (κ cos (ϕn − µ ) + jrn q cos (ϕn − ϑ)) d ϕn 2 I 0 κ 2 − ( rn q ) − j 2rn q cos (µ − ϑ) I 0 (κ ) . (9) Substituting (9) in (8), the exact pdf of the envelope with the elementary phase being the von Mises distribution is given by f R (r ) = ×I 0 ( ∞ N r I0 ∏ ∫ (κ ) ∫ 0 n =1 ∞ 0 f Rn (rn ) ) 2 κ 2 − (rn q ) − j 2rn q cos (µ − ϑ) drn J 0 (qr )qdq (10) Note that (10) shows that the fading channel under some certain conditions may have a complex envelope probability. Therefore, the combining signals in diversity detection should have a mechanism to take the statistics of the imaginary part and phase of the pdf into consideration so as to optimally devise the detection outcome. NUMERICAL ANALYSIS In this section, some examples of the complex pdf are provided to validate the results. Figure 3 shows the comparison of the exact envelope pdf by plotting the real part of (10) the concentration parameter κ as a parameter, mean direction µ = π 4 and the number of fading channels N = 8 , respectively. As expect, when the concentration parameter decreases to zero (i.e., uniform distribution), the exact envelope pdf (real part of complex pdf in (10)) is exact Rayleigh regardless of the mean direction or center of the von Mises pdf. Also for the case of symmetrical pdf, the figure 4 is plotted with the mean direction µ as a parameter, concentration parameter κ = 0.2 and the number of fading channels N = 8 , respectively. As the mean direction µ reduces to zero (i.e., symmetrical about zero), the exact envelope pdf is characterized as the Rayleigh distribution regardless of the density of the elementary phase and envelope f Rn (rn ) . 5. CONCLUSION The complex pdf that is generally used in quantum physics is investigated. It can be shown that for some specific phenomena, the complex pdf can be occurred to model the envelope distribution. This complex pdf has some useful meanings in explaining the characteristics of multipath fading channels in terms of envelope pdf and correlation. The example of elementary phase that is widely used in characterizing the angle of arrival and also has other distribution as special cases is given. Then the exact envelope pdf are derived. To validate the results, some graphical presentations are provided and compared with some specific cases of Rayleigh distribution. Numerical results presented showed that the complex pdf can provide a useful tool for performance analysis. REFERENCES [1] M. F. Pop and N. C. Beaulieu, “Limitations of sum-of-sinusoids fading channel simulations”, IEEE Trans. Commun., vol. 49, p. 699-708, October 2001. [2] H. Barkay and N. Moiseyev, “Complex density probability in non-Hermitian quantum mechanics: Interpretation and a formulary for resonant tunneling probability amplitude”, Physical Review A., vol. 64, 2001. [3] Abdullaev, B., Musakhanov, M. and Nakamura, N., “Complex diffusion Monte-Carlo method: tests by the simulation of 2D electron in magnetic field and 2D fermions-anyons in parabolic well”, Proceedings of Quantum Monte Carlo Meeting, p. July 2001. IEEE Trans. Commun., vol. 45, p. 110-118, January 1997. [12] P. Hou, , B. J. Belzer, and T. R Fischer, “Shaping gain of the partially coherent additive white Gaussian noise channel”, IEEE Commun. Lett., vol. 6, p. 175-177, May 2002. [13]M. Abramowitz, and I. A. Stegun, , Handbook of Mathematical Functions, Dover Publications, New York, 1970. 0.7 µ =π 4 N =8 0.6 [5] 0.5 0.4 0.3 0.2 M. Zak, “Incompatible stochastic processes and complex probabilities”, Physics Letters A, p. 1-7, January 1998. [6] A. Abdi, H. Hashemi, and S. NaderEsfahani, “On the pdf of the sum of random vectors”, IEEE Trans. Commun., vol. 48, p. 7-12, January 2000. = 0.3 = 0.2 = 0.1 = 0, Rayleigh fR(r) [4] M. Lebherz, W. Wiesbeck and W, Krank, “A versatile wave propagation model for the VHF/UHF range considering threedimensional terrain”, IEEE Trans. Antennas Propagat., vol. 40, p. 11211131, October 1992. κ 0.1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 r Fig.3: Comparison of the envelope pdf of the multipath fading channel when the elementary phase is von Mises distributed with concentration parameter κ as a parameter. κ = 0.2 N = 8 κ = 0.2 N = 8 0.8 [7] P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves From Rough Surfaces, Pergamon Press Ltd., 1963. κ = 0.2 N =8 0.7 0.6 µ =π 2 =π 3 [8] [9] P. S. Neelakanta, and D. De Groff, Neural Network Modeling: Statistical Mechanics and Cybernetic Perspectives, CRC Press, Boca Raton, FL, 1994. Mardia, K. V. and Jupp, P. E., Directional Statistics, John Wiley & Sons, Ltd., 2000. [10] Abdi, A., Barger, J. A. and Kaveh, M., “A parametric model for the distribution of the angle of arrival and the associated correlation function and power spectrum at the mobile station”, IEEE Trans. Veh. Technol., vol. 51, p. 425-434, May 2002. [11] T. Eng, and L. B. Milstein, “Partially coherent DS-SS Performance in Frequency Selective Multipath Fading”, fR(r) 0.5 =π 4 = 0, Rayleigh 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 3.5 r Fig.4: Comparison of the envelope pdf of the multipath fading channel when the elementary phase is von Mises distributed with mean direction µ as a parameter. 4