Wireless mobile macrodiversity (MaD) radio system with switch and stay (SSC) receiver and two microdiversity (MiD) selection combining (SC) branches operating over Gamma shadowed Kappa-Mu (k-μ) multipath fading environment is considered. Novel, one-folded integral expression for average level crossing rate (LCR) of MaD SSC receiver output signal envelope is obtained. Numerical results of the proposed model are presented and discussed in relation to the system model parameters

Stefanović Časlav

Đošić Danijel

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Information technologies UNIVERSITY THOUGHT Publication in Natural Sciences, Vol. 6, No 2, 2016, pp. 32-37.  doi:10.5937/univtho6-12527 Original Scientific Paper  THE LCR OF WIRELESS MACRODIVERSITY SSC RECEIVER IN THE PRESENCE OF GAMMA SHADOWED KAPPA-MU FADING Časlav Stefanović1, Danijel Đošić2  1Faculty of Natural Sciences and Mathematics, University of Priština, Kosovska Mitrovicа, Serbia. ABSTRACT Wireless mobile macrodiversity (MaD) radio system with switch and stay (SSC) receiver and two microdiversity (MiD) selection combining (SC) branches operating over Gamma shadowed Kappa-Mu (k-µ) multipath fading environment is considered. Novel, one-folded integral expression for average level crossing rate (LCR) of MaD SSC receiver output signal envelope is obtained. Numerical results of the proposed model are presented and discussed in relation to the system model parameters. Key words: LCR, MaD, MiD, multipath, shadowing.  1. INTRODUCTION Macrodiversity (MaD) system consisting of MaD receiver and two or more microdiversity (MiD) branches is often proposed in the literature ((Shankar, 2009), (Mukherjee & Avidor, 2003), (Basnayaka et al., 2013). Moreover, multipath fading can be mitigated by MiD receivers at single BSs (base stations) while shadowing can be mitigated by MaD receiver combining signals from two or more BSs. It is important to note that multipath fading is caused by physical phenomena such as reflection, refraction and scattering of radio waves while shadowing is caused by obstacles between transmitter and receiver. MaD switch and stay (SSC) system with MiD selection combining (SC) as well as MaD SC system with MiD maximal ratio combining (MRC) or SC can be efficiently applied in composite fading environment to improve performances of wireless communication system as shown in((Stefanović, 2015), (Djosic et al., 2016), (Stefanović et al., 2016), (Panić et al., 2011), (Stamenović et al., 2014)). The Kappa-Mu (κ-µ) distribution describes signal envelope variation in line-of-sight (LoS) multipath fading channels when signals propagate in the environment with two or more clusters. Parameter k is known as Rician factor and can be calculated as the ratio of dominant components power and scattering components power while parameter µ is related to the number of clusters in the propagation environment. Moreover, kappa-mu is general distribution, which means that for different values of κ and µ, Rayleigh, Rice and Nakagami-m distributions can be derived. Moreover, this distribution fits well with experimental data and is often applied in multipath environment ((Yacoub, 2007), (Paris, 2014), (Stefanović et al., 2015)). Log-normal distribution and Gamma distribution are used for the purpose of describing shadowing in wireless communication channel.  In this paper, MaD system has been modeled with switch and stay (SSC) receiver at macro level and two dual branch selection combining (SC) at micro level, since its relatively low implementation complexity. Morover, SC and SSC are often considered diversity techniques ((Zhang et al., 2014), (Milic et al., 2016), (Stefanovic et al., 2013), (Zhao et al., 2015)). Gamma distribution is proposed, with the tendency of obtaining closed form solutions due to its mathematical tractability. Moreover, the average level crossing rate (LCR) of MaD SSC receiver output signal envelope in the presence of Gamma shadowed k-µ multipath fading is efficiently calculated in the form of the sum of one-folded integral. Numerical results are presented graphically and the effect of Rician factor, multipath fading severity parameter, shadowing severity parameter, correlation coefficient and threshold of SSC on the proposed MaD model are examined.  32 Information technologies 2. SYSTEM MODEL The MaD system with two MiD SC parts under the influence of Gamma shadowed k-µ multipath fading as shown on Fig. 1. is considered. Operation is the following: when signal envelope average power at the inputs of the first SC, Ω1 is higher than a specified threshold, ΩT, MaD SSC selects the first SC to process the signal to the receiver. Contrary, when Ω1 is lower than ΩT, SSC selects the second MiD SC to provide the signal path. At the micro level, MiD SC selects the branch with the highest signal envelope power to provide the signal to receiver. Input signal envelopes for the first MiD path are denoted with 11y  and 12y , while the second MiD input signal envelopes are denoted with 21y  and 22y . Further, the output signal envelopes of the first and second MiD combiners are 1y  and 2y , respectively, while signal envelope at the output of MaD SSC is denoted with y. SCSCSSCY11Y21Y22Y12 1 2Y1Y2Y Fig. 1 The MaD system with MaD SSC structure and two MiD SC parts. Probability density function (PDF) of κ-µ signal envelope at the inputs of the first SC are respectively (Panic et al., 2013): 1211121 1 11 112 21( 1) 2 ( 1)( 1)2, .( )1, 2iiy i ikkyk kkp y y Ik ee i  (1) where, κ is Rice factor, µ is fading severity factor, Ω1 is related to the local mean power of 1iy  and  I   is the modified Bessel function of the first kind and order   (Stefanovic et al., 2013). The  I   can be transformed by utilization (Gradshteyn & Ryzhik, 2000), so that the CDF expression of k-µ signal envelope at the inputs of the first SC can be expressed as (Stüber, 1996): 11 1121 1 12 2012 11112112 ( 1)( )( 1) 12 ( 1)1 ( 1)( ), .! ( )ii iyy i ykj jjikF y p t dtk ek kkkj yj jΩΩΩΩ      (2) where Ω1 is signal envelope average power at input of first micro-diversity SC and ( , )m x  is incomplete Gamma function. CDF of the first SC output signal envelope is: 1 11 121 1122121 1 1 1 12 212 111 1 1 121 111 12 ( 1)( 1) 1 12 ( 1) ! ( )(( ) ( )1) 1),)((y y yki iiikF y F y F yk ek kk i ik k ki yΩΩΩΩ Ω222 1212 12 2 11 1 ( 1), .2 ( 1) ! ( )iiki yk i iΩΩ (3) Similarly, CDF of SC output k-µ random variable 2y  is: 2 21 221 1122122 2 2 1 12 222 121 2 1 121 212 22 ( 1)( ) ( ) ( )( 1) 1 12 ( 1) ! ( )( 1) ( 1),y y yki iiikF y F y F yk ek kk i ik k ki yΩΩΩΩ Ω222 1222 22 2 21 1 ( 1), .2 ( 1) ! ( )iiki yk i iΩΩ            (4) Signal envelope average power at inputs of two MiD SC structures, Ω1 and Ω2 are Gamma distributed ((Xekalaki et al, 2003), (Yue et al., 2001)):  33 Information technologies 1 2201 21 2201 2(1 )1 21 2 2 1 10121 1 2202 12 1 1 200 0 (1 )1 11 2( )(1 )2(1 )1( )(1 ) (1 )1! (( )).)(cc cci cc cii c i cp ecIcei i cΩ ΩΩ(5) where ρ is correlation coefficient of the shadowing process, c is shadowing parameter and  Ω0 is mean value of Ω1 or Ω2.  3. LCR OF MAD SSC RECEIVER OUTPUT SIGNAL LCR of Kappa-Mu random variable 1iy  is (Panic et al., 2003): 1 111211111012 12 211 12 21( 1) 2 2 111111)2 2 ( 1) ( 1)1,  1, 2.! (()i iiiy y iimikkyiiy i iN p y df k kykk ey e ii iy yΩΩΩ (6) where 11 11iiy iy ip y y  is joint probability density function (JPDF) of 1iy  and its derivative, mf  is maximal Doppler frequency. Similarly, LCR of random variable 2iy  is: 121221 212 12 211 22 22( 1) 2 2 121 12 2 ( 1) ( 1)1,  1, 2.! ( )iiimyikkyiif k k kNk ey e ii iΩ(7) Since the fading is identical and independent, JPDF of the MiD SC output signal envelope 1y  and its first derivative is: 1 1 11 11 12 12 12 1111 11 121 1 1 1 1 1 11 1 1( ) ( ) ( ) ( )2 ( ) ( ).y y y y y y y yy y yp y y p y y F p y y F yp y y F y(8) The LCR of the first MiD SC output signal envelope 1y  is: 1 1 1 12 1111211 1221 1 1 1 1012 12 211 12 211( 1)2 2 2 11 1 11 1 2 2121 12 (2 2 ( 1) ( 1)1 ( 1)! (( ) ))( 1)y y y y yimikkyikiiN y p y y d y F y Nf k k kk ekx ei ik ek kΩΩΩΩΩ21122 12 12( 1)1 ( 1), .! ( )ikki yi iΩΩ  (9) The average level crossing rate of the second micro-diversity SC output signal is:  121221 22 2212 12 211 12 221( 1)2 2 2 12 1 11 1 2 222 121 22 22 2 ( 1) ( 1)1 ( 1)! ( )( 1)( 1)1! (imxikkyiki iif k k kNk ekx ei ik ek kki iΩΩΩΩΩΩ22 22( 1), .)ki yΩ(10) LCR of MaD SSC output signal envelope is equal to LCR of the first MiD SC output signal envelope when the first MiD SC provides signal to mobile user and the total power at its inputs is higher than predetermined threshold or when the second MiD SC receiver provides signal to the mobile user and the total power at its inputs is lower than the threshold. On the other hand, LCR of MaD SSC output signal envelope is equal to LCR of the second MiD SC output signal envelope when the second MiD SC provides the signal to mobile user and the total power at its inputs is higher than the predetermined threshold or when the first MiD SC provides the mobile user and the total power at its input is lower than the predetermined threshold. Accordingly, the LCR of MaD SSC output signal envelope is: 34 Information technologies 1 1 2TT2 1 22 1 21 1 21 1 2T2 1 2x 1 y| 1 2 201 y| 1 2 20 02 | 1 2 102 | 1 2 10 01 | 1 2 202 y| 1 2 10 0( )( )(1N d  N p d21d  N p d2)( )(1212d  N p d( I))TTTyyyd N p dd N p dd N p dΩ Ω ΩΩΩΩ Ω ΩΩ Ω ΩΩΩΩ Ω ΩΩ Ω ΩΩΩΩ Ω ΩΩ Ω Ω ΩΩ Ω Ω ΩΩ Ω Ω ΩΩ Ω Ω ΩΩ Ω Ω ΩΩ Ω Ω Ω 1 2 .I(11) where1|yN Ω and 2y|N Ω LCR at the first and second input of SSC obtained in (9) and (10), respectively and 1 2 1 2pΩ Ω ΩΩ  is Gamma distribution already given in (5).  The LCR of MaD system with MaD SSC receiver and two MiD SC branches can be obtained by solving the following integrals (Gradshteyn & Ryzhik, 2000): 1 1 211212231 1 2 1 1 203 12 2 2 11 202 12 2 101 12 1 12 2 020 0( ) ( )( )( )( )( ) ( |2 2 11  1y 1! ( )1 1 1! ( ) 11)(()) 1Tyjmkjjjjjc cjI d d N y pf kk kk ek kj jj j k cΩ ΩΩΩ Ω Ω Ω ΩΩΩ333 12 121 02 1203 3122( )(3 1 211  11)1( 1 )!Γ( )1())( )(.,Tj ci cj c jkyj j cki c j ye dΩΩΩ ΩΩΩΩΩ(12) Integral 2I  is (Gradshteyn & Ryzhik, 2000): 1 1 2112 2 1 1 1 20 03 12 2 2 11 20( ) (|2 2 ( 1)( 1))TxjmkjI d d N x pf kk kk eΩΩ ΩΩ Ω Ω Ω Ω 212233332 12 2 101 12 1 12 2 02 12020 3 303 1201 y ( 1)! ( )1 1 1! ( ) ( 1) ( ) 11( 1 )!Γ(( ))1,1( )( )( )Tjjjjc cj ci cjj cTk kj jj j k cj j ci cΩΩΩΩΩΩΩ12 121 012( 1)  12 ( )2 11( 1), .jkykj y e dΩΩ Ω ΩΩ(13) 4. NUMERICAL RESULTS Average level crossing rate (LCR) is important second order performance measure of wireless communication system, which determines the number of signal threshold crossings in positive going direction. -10 0 101E-30.010.11  LCR*fmy[dB]  k=1,  c = 1,T=0.5  k=1,  c = 1,T=0.5  k=1,  c = 1,T=0.5  k=2,  c = 1,T=0.5  k=3,  c = 1,T=0.5  k=1,  c = 2,T=0.5  k=1,  c = 3,T=0.5Fig. 2. Normalized LCR of MaD SSC receiver for several values of parameter mu, kappa and c and constant values of ρ, Ω0, and Ω T. Figure 2. presents normalized LCR of MaD SSC system for various values of fading severity parameters µ, Rice factor k shadowing severity c, and constant values of correlation parameter ρ, average power Ω0 and threshold value ΩT. It is evident that LCR increases for lower values of y, reaches its maximum, and then decreases for higher values of y. It can been seen that by increasing parameter µ and k, it comes to the improvement of the performances of MaD SSC system, since LCR decreases. For lower values of y, LCR decreases as parameter c increases while for higher values of y, LCR increases as parameter c increases. Moreover, parameter c has greater impact on LCR then parameter µ and k. 35 Information technologies -10 0 101E-30.010.11 k=1,  c=1,T=0.5 k=1, c=1, T=0.5 k=1, c=1, T=0.5 k=1, c=1, T=0.5  LCR*fmy[dB] Fig. 3. Normalized LCR of MaD SSC receiver for constant values of mu, kappa, c, Ω0 and ΩT and several values of ρ. -10 0 101E-30.010.11 k=1,  c=1,T=0.2 k=1, c=1, T=0.3 k=1, c=1, T=0.5 k=1, c=1, T=0.8  LCR*fmy[dB] Fig. 4. Normalized LCR of MaD SSC receiver for constant values of mu, kappa, c, Ω0 and ρ and several values of ΩT.  Fig. 3. shows normalized average level crossing rate versus y for various values of ρ and constant values of µ, k, c, Ω0 and ΩT. Since there is no correlation for ρ=0, the impact of ρ on LCR can be noticed for higher values of ρ.  Fig. 4. shows normalized average level crossing rate versus y for various values of ΩT and constant values of µ, k,  c,  Ω0 and ρ.  5. CONCLUSION In this paper, MaD technique with MaD switch and stay combining reception and two MiD selection combiners in correlated Gamma shadowed Kappa-Mu multipath fading channel is considered. MaD system with MaD SSC receiver is proposed, since it has lower complexity implementation then other diversity techniques. LCR of SSC MaD receiver output signal envelope in the form the sum of one-folded integrals is obtained and graphically presented in relation to different system model parameters.  REFERENCES Basnayaka, D.A., Smith, P.J., & Martin, P.A. 2013. 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The LCR of wireless macrodiversity SSC receiver in the presence of gamma shadowed Kappa-Mu fading

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The LCR of wireless macrodiversity SSC receiver in the presence of gamma shadowed Kappa-Mu fading

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