Microsoft Word UK VN TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ, ĐẠI HỌC ĐÀ NẴNG SỐ ���� 1 ĐẶC TRƯNG THỐNG KÊ CỦA CÁC MÔ HÌNH FADING RAYLEIGH TRONG THÔNG TIN VÔ TUYẾN STATISTICAL PROPERTIES OF RAYLEIGH FADING MODE[.]
TẠP CHÍ KHOA HỌC VÀ CƠNG NGHỆ, ĐẠI HỌC ĐÀ NẴNG - SỐ ĐẶC TRƯNG THỐNG KÊ CỦA CÁC MƠ HÌNH FADING RAYLEIGH TRONG THƠNG TIN VƠ TUYẾN STATISTICAL PROPERTIES OF RAYLEIGH FADING MODELS IN WIRELESS COMMUNICATIONS Vien Nguyen-Duy-Nhat, Hung Nguyen-Le, and Chien Tang-Tan Department of Electronics and Telecommunications Engineering, Danang University of Technology E-mail: ndnvien@dut.edu.vn, nlhung@dut.udn.vn, ttchien@ac.udn.vn ABSTRACT The paper studies statistical properties of existing fading channel models in wireless communications Several statistical characteristics of various fading channel models are numerically verified with related theoretical values The comparison results can be used as guidelines in selecting suitable fading channel models for a specific wireless system design Keywords: Rayleigh fading channels, fading simulator, sum-of-sinusoids channel simulator, correlation properties Introduction The past decade has witnessed explosive growth in wireless data traffic of ubiquitous multimedia services for both business and residential customers worldwide To meet the development trends, numerous advanced transmission techniques have been proposed for wireless communications Unlike wireline transmissions, wireless communications lends itself to many technical challenges in signal processing due to the randomness of radio channel responses As a result, understanding the nature of wireless channels is of paramount importance in optimizing the wireless system performance under limited radio resources and power constraints To characterize the complex nature of wireless channels, several mathematical models have been proposed in the literature [1-8][12] Among them, an early fading channel model was proposed by Clarke [1] Jakes [2] has proposed a simplified version of Clarke’s model for generating multiple Rayleigh fading waveforms deterministically by superimposing multiple sinusoidal, are called sum-of-sinusoids (SoS), with different frequencies and initial phases The simplified model has been widely accepted for about three decades Later, Dent [3] modified Jakes’ model by using orthogonal Walsh-Hadamard code words to generate the multiple uncorrelated faded envelopes Pop and Beaulieu [4] showed that there are some undesirable properties in the Jakes' model More specifically, the autocorrelation functions (ACF) of the in-phase differs from that of quadrature components, and the crosscorrelation function (XCF) between the inphase and quadrature components is not always zero For any pair of faders, they are not mutually independent because the XCF between them is generally not zero To alleviate the drawbacks, Li and Huang proposed a model [5] based on Jakes’ model, this model overcomes some undesirable properties in the Jakes and Dent models The in-phase and quadrature components in any single fader are independent and have the same autocorrelation functions However, the model of Li and Huang possesses a highly computational complexity To reduce computation complexity, a novel model is proposed by Wu [6] The model’s correlation properties are as good as those of Li and TẠP CHÍ KHOA HỌC VÀ CƠNG NGHỆ, ĐẠI HỌC ĐÀ NẴNG - SỐ Huang model, yet computational complexity is reduced by a half Zheng and Xiao [7-9] proposed a new model and this model have been widely used for Rayleigh fading channels in recent years The SoS models can be classified into statistical and deterministic Deterministic models have fixed phases, amplitudes, and maximum Doppler frequencies for all simulation trials In contrast, the statistical models have at least one of above parameters as a random variable The statistical properties of the models will vary for each simulation trial, but converge to the desired properties with a large number of simulation trials An ergodic statistical model converges to the desired properties in only a simulation trial The simulation model must ensure accurately evaluation under realistic fading conditions In Rayleigh fading simulation model, the in-phase and quadrature components of the complex Gaussian Widesense stationary (WSS) process = + , and , are independent, zero mean Gaussian with equal variance The envelope || = + follows the Rayleigh distribution || = , where denotes the time-averaged power of the fading process at the receiver The ideal auto-correlation functions of the in-phase or quadrature parts and the complex envelope are scale with the zeroth-order Bessel function of the maximum Doppler frequency and the time lag The ideal cross-correlation function between the quadrature and in-phase components is zero In this paper, some important statistical properties of the SoS fading simulation models are analyzed and compared with each others The numerical results show that the model with correct statistical properties for Rayleigh fading channel simulation The rest of this article is organized as follows Section reviews the Clarke’s reference SoS fading models with its desired statistical properties The SoS based Rayleigh fading simulation models are described in Section The statistical properties of SoS Rayleigh models are simulated and evaluated in Section Finally, Section concludes the paper 2.1 Clarke’s Reference Model Clarke’s Reference Model Clarke [1] showed that the complex channel envelope at the time t can be expressed by = ∑& '( !"#$" , (1) where )* = 2, * is the maximum argular Doppler frequency, * is the maximum Doppler frequency, * = - with /* is the maximum vehicle speed, is the RF carrier frequency and is the speed of light 1, , 2 and 3 are the number of propagation path, the path gain, the angle of arival, and the phase associated with the 4 5 propagation path, respectly Assuming that , 2 , and 3 are mutually independent 2 and 3 are uniform distributed on interval 6−,,8, 2.2 Statistical Properties According to the Central Limit Theorem, the in-phase and quadrature components of the complex faded envelope are Gaussian random processes for large N Thus, the envelope || is Rayleigh distributed and @ B the phase Θ = argtan ?@ A BD is uniformly distributed C The statistical properties of the Clarke’s model can be consulted in [2] and [10] for the autocorrelations and cross-correlations of the reference model are summarized by EF,G ,F,G H = IJ, + H, K = LM )* H (2) TẠP CHÍ KHOA HỌC VÀ CƠNG NGHỆ, ĐẠI HỌC ĐÀ NẴNG - SỐ EF,G H = IJ + H K = 0, EFF H = (3) 2L ) H, U = V , EO ,P H = I6Q + HR S = T M * , U ≠ V (4) EF,G G,F H = l∑a#( '( j E H = (5) 3.1 [ & Sos-Based Rayleigh Fading Simulation Models Jake’s Model ( From (1) and selecting = & , Z = , and 3 = for = 0, , 1, Jake [2] derived deterministic simulation model for Rayleigh fading channels The complex faded envelope as = + = = √& √& (6) ∑a#( '( ] cos )* + 3 ∑a#( '( b cos )* + 3 (7) 2cde , = 1,2, , g8 √2cde , = g + 2dh4e , = 1,2, , g8 b = T √2dh4e , = g + [ , = 1,2, , g e = i &[ ) = k j , = g + )* cd2 [ & , = 1,2, , g8 )* , = g + j & o" cos) Hn m" o" (13) (14) cos ) Hn (15) p & 3.2 (q&( & (17) Pop and Beaulieu’s Model Pop and Beaulieu improved Jakes’ fading channel simulator to eliminate the stationary problem occurring in Jakes’ original design Pop and Beaulieu modified Jakes’ model by removing the constrain 3 = and allowing 3 to be independent random variables uniformly distributed over 6−,8, 8, for all n = 1, 2, , N When approaching infinity, the ACF and XCF of the in-phase and quadrature components, the envelope and the squared envelope of fading signal are given by [11] EFF H = L )s H + Lj )* H (19) E H = 2LM )* H (21) [/ sin4Z cos )* H cos θ wZ (20) E|||| H = 4LM )* H + 4Lj )* H [/ +4 l tM [ 3.3 (18) EGG H = L )s H − Lj )* H EF,G G,F H = tM [ (11) The autocorrelation and crosscorrelation functions of the quadrature components, and the autocorrelation functions of the envelope and the squared envelope of fading signal are given by [11] cos ) Hn 62 ∑a#( '( cos) H + cos )* HS (16) (10) (12) & +4E F G H + LM 2)* H + (8) (9) m" E|||| H = + 2E FF H + 2E GG H where N=4M=2, 3 = 0 for all n, and ] = T j & where E[.] is the expectation operator, LM is the zero-order Bessel function of the first kind l∑a#( '( EGG H = l∑a#( '( E|O |,|O | H = I6|Q + H| |Q | S = + 4LM )* H j & sin4Z cos )* H cosZ wZn (22) Dent’s Model To overcome the correlation shortcoming of the Jake’s model, Dent [3] suggested that the U 5 complex faded envelope can be generated by using TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ, ĐẠI HỌC ĐÀ NẴNG - SỐ &y Q = ∑'( zQ 46cde cd)* cd2 + &y 8 (23) Z +dh4e cd)* cd2 + Z S where U = 1,2, , 1M ; 2 = [[ ; & [ ; &y e = Z are the independent random phases, each of which is uniformly distributed in 608, 82,; Ak(n) is the U 5 Walsh-Hadamard orthogonal codeword to decor relate the multiple faded envelopes in Jake’s model When Ak(n) is chosen as the Walsh– Hadamard function set, the statistical properties of Dent’s model are given by [5] EFF H = EGG H = ∑&y cos e cos) cos2 H(24) &y '( ( ( &y EF,G G,F H = y ∑& '( sin e cos) cos2 H (25) ( &y y ∑& '( sin2e cos) cos2 H (26) EO P H = ( y ∑| }'( zQ 4zR 4cd) cd2 H (27) &y 3.4 Li and Huang’s Models The independence between different faded envelopes in the Dent’s model is still not so good, so Li and Huang [5] have proposed a novel model as the following &y ( 6cd)a cd2Q + ZQ 8 Q = ~ 1M 'M 8+ dh4)a dh42Q + Z S Q (28) Where U = 1, , , ZQ are independent random phases uniformly distributed on the interval 60; 2,, 2Q is the 4 5 angle of arrival in the U 5 complex faded envelope, )* is the maximum angular Doppler frequency The angles of arrivals are 2Q = Q[ [ + & & for = 0, , g, where 2MM is an initial angle of arrival, chosen to be < 2MM < 2MM ≠ ,/1 [ & and The autocorrelation function of the quadrature component of the faded envelope, the cross-correlation function between the inphase and the quadrature components and the cross-correlation function between the inphase and the quadrature components are then derived as [5] y EFF H = 2 ∑'M cos) cos2Q H (29) & ( y EGG H = 2 ∑'M cos) sin2Q H (30) & ( EF,G G,F H = (31) &y ( cd) cd2Q H + EP P H = 2 ∑'M cd) dh42Q H (32) 3.5 Zheng and Xiao’s Model Zheng and Xiao proposed several novel statistical models [7-9] by allowing all three parameter sets (amplitudes, phases, and Doppler frequencies) to be random variables Q = ∑a '( cdQ cd)a cd2Q + 3Q a 8+ ∑a '( dh4Q cd)a cd2Q + 3Q where 2Q = [[#O ,4 ja , (33) = 1,2, , g; ZQ , Q , 3Q are statistical independent random variables uniformly distributed on the interval 608, 82, for all n and k The autocorrelation and crosscorrelation functions of the in-phase and quadrature components, the envelope and the squared envelope of faded envelopes are given by [7] when g → ∞ EF,G ,F,G H = IJ, + H, K = LM )* H(34) EF,G H = IJ + H K = 0, (35) EO ,P H = I6Q + HR S = T 2LM )* H, U = V , , U ≠ V (36) E|O |,|O | H = I6|Q + H| |Q | S TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ, ĐẠI HỌC ĐÀ NẴNG - SỐ = + 4LM )* H Zajic and Stuber’s Model Zajic and Stuber proposed an new ergodic statistical model The U 5 complex faded envelope is defined as Q = Q + Q , where coseQ cos)* cd2Q + Q = ∑a √& '( (38) 3Q , sineQ sin)* cd2Q + Q = ∑a √& '( 3Q (39) It is assumed that P independent complex envelopes are desired U = 0, , , each having g = & j 0, , − + 4.1 Fig show the PDFs of the generated SoS Rayleigh fading models which are plotted and compared with PDF of Rayleigh distribution (with variance=1) when the number of random trials is 10 We can see that the models of Dent [3], Li [5] and Zheng [7] are in good agreement with the PDF of theoretical values of Rayleigh distribution while the others not [ , & for = 1, , g; U = PDFs of the faded envelope 0.8 0.7 EF,G H = 0, 2L ) H, U = V , EO ,P H = T M * , U ≠ V 0.6 0.5 0.4 0.3 0.2 0.1 0 50 Figure 100 150 200 c(t) 250 300 350 400 The PDFs of faded waveforms PDFs of the faded envelope (40) Theoretic N=10 N=50 N=100 0.9 0.8 (41) 0.7 (42) E|O |,|O | H = + 4LM )* H, g → ∞ (43) Performance Evaluation The performance evaluation of the SoS fading simulation models was carried out by comparing the statistical properties with each others A computer simulation was implemented to generate SoS complex faded envelopes using 1M = 8 oscillators With a Doppler frequency * = 20 , sampling PDF of c(t) EF,G ,F,G H = LM )* H Theoretic Jake Dent Li Pop Zheng Zajic 0.9 When g → ∞, derivation of the autocorrelation function of the in-phase and quadrature components, auto-correlation function of the squared envelope is presented by [12] Evaluation of PDFs of the Envelope and Phase sinusoidal terms in the I and Q components The parameters 3Q , eQ and Z are independent random variables uniformly distributed on the interval 6−,, , The angles of arrivals are chosen as follows: [ [Q 2Q = & + & frequency = 1 The ensemble averages for all the simulation results are based on 10, 50, and 100 random trials as indicated in the figures PDF of c(t) 3.6 (37) 0.6 0.5 0.4 0.3 0.2 0.1 0 50 100 150 200 250 300 350 c(t) Figure Envelope PDF of the waveform generated by the Zheng model [7] for various numbers of trial simulation Moreover, as observed from Fig 2, an increase in the number of trials results in a TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ, ĐẠI HỌC ĐÀ NẴNG - SỐ Evaluation of Correlation Statistics The simulation results of the autocorrelations of the quadrature components, the cross correlations of the quadrature components, and the autocorrelations of the complex envelope and squared envelope of the simulator output are shown in Figs 3–6, respectively The second-order statistics of the mathematical ideal model, which are analyzed above, are also included in the figures for comparison purposes Figs 1-3 show the autocorrelations of the complex envelope, the cross-correlation function between the inphase and the quadrature components, and the autocorrelations of squared envelope for these models with the number of trials is 10 Theoretic Jake Dent Li Pop Zheng Zajic ACFs of the complex envelope 1.5 Theoretic Jake Dent Li Pop Zheng Zajic 0.6 0.4 0.2 -0.2 -0.4 50 100 150 200 t(Second) 250 300 350 Figure The cross-correlation function between the in-phase and the quadrature components 1.2 Theoretic Jake Dent Li Pop Zheng Zajic ACFs of the in-phase component 4.2 XCFs of the in-phase and quadrature components better agreement with the theoretical ones 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.6 0.5 -0.8 20 40 60 80 100 120 t(Second) 140 160 180 200 Figure The auto-correlation of I/Q-components -0.5 -1 50 100 150 200 t(Second) 250 300 350 Figure The auto-correlation of faded waveforms From Fig 3, we can observe that the auto-correlation of the complex envelope of Pop’s [4] and Zheng’s [7] models are identical with the theoretical ones of the reference model In contrast, the models of Dent and Li are different from the theoretical ones Other comparisons are summarized in Fig 4, where we plot the cross-correlation of the real and imaginary parts of the faded waveforms The models of Zheng [7] and Zajic [12] agree very well with the theoretical autocorrelation given by (3) In Fig we compare the ACFs of the in-phase component for the reference model with the other simulation models The models of Zheng [7] and Zajic [12] have the best uncorrelation property Compared with the others models, the Zheng’s [7] and Zajic’s [12] models provide similar approximations to the theoretical ACFs of the squared envelope as in Fig.6, where the squared envelope correlation is plotted for different time values TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ, ĐẠI HỌC ĐÀ NẴNG - SỐ 0.45 ACFs of squared envelope 6.5 MSE of the ACFs and XCFs Theoretic Jake Dent Li Pop Zheng Zajic 7.5 5.5 4.5 MSE of PDF MSE of Rc(t) 0.35 MSE of Rc (t) i MSE of Rc (t) q 0.3 MSE of R|c(t)|2 MSE of Rc (t)c (t) i q 0.25 MSE of Rc (t)c (t) l k 0.2 0.15 0.1 0.05 3.5 0.4 20 40 60 80 100 120 t(Second) 140 160 180 200 Models: Jake; Dent; Li; Pop; Zheng; 6.Zajic Figure The auto-correlation of faded squared waveforms Figure The MSE of statistical properties with M=8, N=100 The accuracy of the SoS fading simulation models can be measured by the mean-square-error (MSE) Figs 7, and Table I summarize MSEs of PDF, ACF of the complex envelope, the in-phase and quardrature components, the squared envelope, and the XCFs of the intra and inter waveforms The Fig.7 and Table show numerical results when the number of simulation trials is 10 The MSE of statistical properties using 100 trials are presented in the Fig We can observe that the Zheng’s [7] and Zajic’s [12] models have the best statistical properties But when increase the number of trials, the Zajic’s [12] model achieves a larger de-correlation than the Zheng’s [7] and the other models 0.35 0.3 MSE of PDF MSE of Rc(t) MSE of the ACFs and XCFs MSE of Rc (t) i 0.25 MSE of Rc (t) q MSE of R|c(t)|2 0.2 0.15 MSE of Rc (t)c (t) i q MSE of Rc (t)c (t) l k Table1 Mean-Square-Error of Correlation Functions Complex Envelope In-phase component Quadrature component MSE of XCF Jake 0.0087 0.0181 0.0225 0.0127 Dent 0.0255 0.0482 0.0451 0.0238 0.0135 0.0191 0.0217 0.0154 0.0091 0.0185 0.0237 0.0126 0.0012 0.0047 0.0033 0.0047 0.0022 0.0029 0.0049 0.0030 MSE of ACF SoS model Li and Huang Pop and Beaulieu Zheng and Xiao Zajic and Stuber 0.1 0.05 5 Models: Jake; Dent; Li; Pop; Zheng; 6.Zajic Figure The MSE of statistical properties with M=8, N=10 Conclusion The paper presented an analysis of various SoS fading simulation models in terms of statistical properties Based on the numerical results, we can conclude that the statistical properties of the fading models proposed by Zheng [7] and Zajic [12] well coincide with the theoretical values We can select one of these models to generate multiple uncorrelated fading waveforms for mobile channels TẠP CHÍ KHOA HỌC VÀ CƠNG NGHỆ, ĐẠI HỌC ĐÀ NẴNG - SỐ References [1] R H Clarke, “A statïstical theory of mobile radio reception,” Bell Systems Technical Journal, 1968 [2] W C Jakes, Microwave Mobile Communications New York: Wiley, 1974 [3] P Dent, G E Bottomley, and T Croft, “Jakes fading model revisited ” Electron Letter, vol 29, no 13, pp 1162–1163, June 1993 [4] M F Pop, and N C Beaulieu, “Limitations of sum-of-sinusoids fading channel simulators,” IEEE Trans Commun, vol 49, pp 699–708, 2011 [5] Y Li, and X Huang, “The simulation of independent Rayleigh faders,” IEEE Trans Commun., vol 50, pp 1503–1514, 2002 [6] Z Wu, “Model of independent Rayleigh faders,” Electronics Letters vol 40, no 15, 2004 [7] Y R Zheng, and C Xiao, “Simulation models with correct statistical properties for Rayleigh fading channels,” IEEE Trans Commun., vol 51, no 6, pp 920-928, Jun 2003 [8] Y R Zheng, and C Xiao, “Improved models for the generation of multiple uncorrelated Rayleigh fading waveforms,” Communications Letters, vol 6, no 6, pp 256–258, Jun 2002 [9] Y R Zheng, and C Xiao, “A statistical simulation model for mobile radio fading channels,” Proc IEEE WCNC’03, New Orleans, USA, pp 144-149, March 2003 [10] G L Stuber, Principles of Mobile Communication, ed.: Norwell, MA: Kluwer, 2001 [11] C Xiao, Y R Zheng, and N C Beaulieu, “Second-Order Statistical Properties of the WSS Jakes’ Fading Channel Simulator,” IEEE Transactions on communications, vol 50, no 6, Jun 2002 [12] A G Zajic, and G L Stuber, “Efficient simulation of Rayleigh fading with enhanced decorrelation properties,” IEEE Transactions on Wireless Communications, vol 5, pp 1866-1875, Jul 2006 ... Zheng Zajic ACFs of the in-phase component 4.2 XCFs of the in-phase and quadrature components better agreement with the theoretical ones 0.8 0.6 0.4 0.2 -0 .2 -0 .4 -0 .6 0.5 -0 .8 20 40 60 80 100 120... Figure The auto-correlation of I/Q-components -0 .5 -1 50 100 150 200 t(Second) 250 300 350 Figure The auto-correlation of faded waveforms From Fig 3, we can observe that the auto-correlation of... the time-averaged power of the fading process at the receiver The ideal auto-correlation functions of the in-phase or quadrature parts and the complex envelope are scale with the zeroth-order