S. Suljovic i dr. Srednji broj osnih presjeka kod SC prijemnika u kanalu s Weibull-ovim fedingom i Gamma sjenkom

ISSN 1330-3651(Print), ISSN 1848-6339 (Online) DOI: 10.17559/TV-20140909142128

LEVEL CROSSING RATE OF SC RECEIVER OVER GAMMA SHADOWED WEIBULL MULTIPATH CHANNEL

Suad Suljović, Dragana Krstić, Srdjan Maričić, Srboljub Zdravković, Vladeta Milenković, Mihajlo Stefanović

Original scientific paper The telecommunication system consisting of selection combining (SC) receiver which works in Gamma shadowed Weibull multiple-faded channel is discussed in this work. The received signal suffers Weibull small scale fading which leads in variation of the signal envelope and Gamma large scale fading which results in variation of the SC receiver output signal envelope power. SC receiver is utilized to abate the impact of Gamma large scale fading effects and Weibull small scale fading effects on system characteristics. The formula for average level crossing rate (LCR) of signal envelope at SC combiner output is performed in the closed shape. The result we get can be applied to calculate the average fade duration (AFD) of such wireless systems. The obtained solutions are plotted in a few graphs to point out the impact of Weibull fading envelope severity parameter and Gamma shadowing severity parameter on system features.

Keywords: Gamma shadowing; level crossing rate (LCR); selection combining (SC); Weibull fading

Srednji broj osnih presjeka kod SC prijemnika u kanalu s Weibull-ovim fedingom i Gamma sjenkom

Izvorni znanstveni članak U ovom radu je razmotren bežični telekomunikacijski sustav sa SC prijemnikom koji radi u prisustvu Weibull-ovog fedinga i Gamma sjenke. Utjecaj Weibull-ovog fedinga na primljeni signal se ogleda u promjeni anvelope signala, a utjecaj Gamma sjenke u promjeni snage izlaznog signala. SC prijemnik se koristi za smanjenje utjecaja fedinga i sjenke na karakteristike sustava. Izraz za srednji broj osnih presjeka izlaznog signala iz SC prijemnika je izveden u zatvorenom obliku. Dobiveni rezultati se mogu koristiti za izračunavanje prosječnog trajanja otkaza bežičnog sustava. Numerički rezultati su predstavljeni grafički, kako bi se prikazao utjecaj parametara fedinga i sjenke na karakteristike sustava.

Ključne riječi: broj osnih presjeka; Gamma sjenka; SC kombiniranje; Weibull-ov feding

1 Introduction signal envelope power in shadowed fading channels by usage of the log-normal distribution is more suitable Multipath fading and shadowing deteriorate because it assures better matching of measurings and characteristics of wireless communication systems and numerical results. On the other side, the usage of Gamma restrict the channel capacity. The signal at the reception is distribution allows much simpler mathematical exposed simultaneously to both large and small scale manipulations and formulas for obtaining all system fadings. The fast fading creates the signal envelope performance. More recent papers have shown that log- variation and slow fading creates the variation of the normal and Gamma distributions get along well [9, 10]. signal envelope power [1 ÷ 3]. The model of signal power described by Gamma There are several distribution which are utilized to distribution makes the analysis of the system easier and delineate variations of signal envelope and variations of gives the expressions for system performance (such as signal envelope power in fading channels. The most probability density function (PDF) of signal to noise ratio commonly used distributions for describing small scale (SNR)) in the closed shape. variations of signal envelope are: Rician, Rayleigh, Diversity techniques are widely applied methods for Weibull, Nakagami-m, Nakagami-q(Hoyt) and α-μ mitigation of the fading influence and improvement of the distributions [4 ÷ 8]. The variation of signal envelope wireless mobile system performance without increasing power is usually depicted by log-normal distribution or the channel bandwidth or power of transmited signals. Gamma distribution [2]. Diversity combiners combine more received signals in Nakagami-m distribution and Rayleigh distribution different ways. Maximum ratio combining (MRC) has the describe short term signal envelope variations in linear, best performance. Selection combining (SC) is the least non line-of-sight multiple-faded environment.Weibull and complicated. In SC receivers, only one of diversity α-μ distributions are used in non linear and non line-of- branches is processed at the receiver at one moment [3]. sight faded medium. In linear and line-of sight channels, It is important to determine the statistical Rician distribution is used for a good description of short characteristics of the wireless system of the first two term signal envelope variation [3]. orders. The first order statistics of a communication Weibull distribution has two parameters: shape system are: the bit error rate (BER), the outage probability parameter α, α > 0, and scale parameter Ω, Ω > 0 (Eq. (Pout) and the channel capacity. The second order (9)). The Weibull distribution is general distribution. Out properties of communication systems are: the joint of it, the exponential distribution can be obtained for α =1 probability density function of the signal envelope and its and the Rayleigh distribution for α = 2 [2]. first derivative at the output, the average level crossing In shadowed fading channels, the long term variation rate of signal envelope at the output and the average fade of signal envelope power can be inscrabed by log-normal duration (AFD) [2]. and Gamma distributions. The modeling of variations of

Tehnički vjesnik 23, 6(2016), 1579-1584 1579 Level crossing rate of SC receiver over gamma shadowed Weibull multipath fading channel S. Suljovic et al.

2 Related works the authors reviewed and compared the results from some previously published papers in a more general way. In scientific literature many articles can be found In later papers the shadowing is taken into account. In which are engaged with the second order properties of such a way, the paper [21] treats second order wireless telecommunication systems working over characteristics of the signal envelope in the receiver in composite shadowed multiple-faded environments [11 ÷ mobile channels under the influence of Weibull 15]. fading and shadowing. The output signal is represented as First of all, in the eighties, the influence of composite the product of Weibull and log-normal process. The Rayleigh fading was considered. The small scale Rayleigh mathematical solutions for LCR and AFD for signal fading acting together with large scale log-normal envelope are derived. The parameters of fading and changes of mean value is analyzed in [11]. The interest shadowing affecting this statistics are shown in several for the investigation of influence of Rayleigh fading in graphs. wireless channels is present even today [12]. In frequency non-selective fading environments, the In the nineties, the log normal-shadowed Rician waves that occur in receiver are not homogeneous channels were scrutinized [13,14]. An estimation of because of nonuniform scattering from the objects of spectrum efficiency of microcellular land mobile radio reflective nature. Besides, large scale fading affects the system is shown in [13] by deliberation the useful signal level of received signal by inducing slow variations to its as fast fading with Rician distribution along the log- mean value. In [22], the authors studied propagation normal slow fading and co-channel interference as taking into account both, inhomogeneous scattering and uncorrelated Rayleigh fading acting together with slow shadowing, by multiplication of the Weibull process and fading with log-normal distribution. Then, the log-normal process. The Weibull one includes scattering microcellular system is likened to a standard nonuniformities of the environment, while log-normal macrocellular system. A new method is applied in [14] to process takes into account the slow term changes of the extend earlier result by involving the effect of log-normal mean value because of shadowing. The correct expression slow fading on the Pout for microcellular radio systems in for the composite PDF is provided as well as approximate Rician/Rician fading environments. expressions for the second order characteristics (LCR and Later, an analysis was done for system with macro ADF). and micro diversity receivers in shadowed fading The level crossing rates of SC receiver output signal environment. The existence of Rician fading and are set by these authors in [23] for two examples: 1) the correlated Gamma shadowing is studied in [15]. The LCR presence of Gamma shadowing and Rician multipath and AFD of macro SC diversity combiner and two micro fading; 2) the presence of Gamma shadowing and k-µ MRC diversity combiners which function in such Gamma multipath fading. In this article, the wireless shadowed Rician multipath fading environment are telecommunication system with one SC combiner computed. performing in Gamma shadowing and multipath fading In [16], macrodiversity system with SC macro channel will be discussed also, but the signal in channel diversity combiner and two microdiversity MRC being affected at the same time by Gamma slow fading combiners working in Gamma shadowed Nakagami-m and Weibull fading. Weibull multipath fading creates fast fading environment is investigated. The macro variations of signal envelope and Gamma slow fading diversity SC combiner minimizes shadowing action on creates variations of signal envelope power. system characteristics and micro diversity MRC The joint PDF of SC combiner output signal envelope combiners mitigate Nakagami-m multipath fading and its first derivative is obtained by using the joint PDF influence to the system properties. The second-order of signals and their first derivatives on the inputs of SC characteristics analysis of this system is presented. combiner. The average LCR of SC combiner output An access with infinite series for determination the signal envelope is deffined as the average value of the system performance of a switched and stay combining first derivative of SC combiner output signal envelope (SSC) receiver with two branches, functioning in a [2]. The SC combiner is used to alleviate the multipath channel with correlated Hoyt fading, was given in [17]. fading and shadowing impacts on system performance. The evaluation is on the basis of the average bit error After obtaining the expression for average LCR of SC probability (ABEP) and the Pout. combiner output signal in the closed form, the results are Easy and precise formulas for characteristics of an depicted in some figures. equal gain combining (EGC) combiner with L branches, By the author’s knowledge, the second order in an independent Hoyt fading environment, are obtained characteristics of wireless telecommunication system with in [18]. By using an approximate but adequate PDF of the SC combiner which acts in Gamma shadowed Weibull sum of random variables with Hoyt distribution, the fading medium is not published in the available scientific formulas for the average BER for binary and both, literature. coherent and noncoherent , are performed. The second order characteristics (average LCR and 3 Level crossing rate AFD) and average Shannon’s channel capacity for channel with Weibull multipath fading are acquired in The samples of Weibull random process can be [19] as formulas in the closed form. In [20], the formulas obtained by nonlinear transformation of samples of for the average LCR and AFD at the output of SC Rayleigh random process. Weibull random variable is [2]: combiner with two branches are obtained. In that paper, = 𝑧𝑧 , (1) 𝛼𝛼 1580 𝑥𝑥 𝑦𝑦 Technical Gazette 23, 6(2016), 1579-1584 S. Suljovic i dr. Srednji broj osnih presjeka kod SC prijemnika u kanalu s Weibull-ovim fedingom i Gamma sjenkom where y is Rayleigh random variable. Squared Rayleigh The LCR of Weibull random envelope shall be random variable is a summation of two squared random computed as an average value of the first derivative of variables with Gaussian distribution [1]: Weibull random envelope [2]:

= = + , (2) ∞ = ( ) = ∝ 𝑧𝑧 2 2 1 2 𝑥𝑥with y𝑦𝑦 and𝑦𝑦 y being𝑦𝑦 independent, zero mean, random 𝑥𝑥 𝑥𝑥𝑥𝑥̇ 1 2 𝑁𝑁 � 𝑑𝑑𝑥𝑥̇ 𝑥𝑥̇𝑃𝑃 𝑥𝑥𝑥𝑥̇ variables with Gaussian distribution and same variances 0 ∞ 1 2 σ . = 𝑥𝑥̇ 2 = 1 𝛼𝛼 2 The first derivative of Weibull random variable is: 𝛼𝛼 𝛼𝛼−1 −�𝛺𝛺𝑥𝑥 � −�2𝜎𝜎𝑥𝑥̇ 2� 𝑥𝑥 𝑒𝑒 �0 𝑑𝑑𝑥𝑥̇ 𝑥𝑥̇ 𝑥𝑥̇ 𝑒𝑒 𝛺𝛺 √ 𝜋𝜋𝜎𝜎 = ( + ). (3) ( ) = 𝛼𝛼 . (11) �/ 1 𝛼𝛼 1 𝑚𝑚 2 ∝−1 𝜋𝜋𝑓𝑓 𝑥𝑥 − 𝛺𝛺𝑥𝑥 𝑥𝑥̇ ∝𝑥𝑥 𝑦𝑦1𝑦𝑦̇1 𝑦𝑦2𝑦𝑦̇2 1 2 The first derivation of random variable with Gaussian √2𝜋𝜋 𝛺𝛺 In Gamma𝑒𝑒 shadowed Weibull multipath fading distribution is random variable with Gaussian distribution. environment, the signal envelope power has Gamma These variables and are random variables with distribution: Gaussian distribution. The random variables with 1 2 𝛺𝛺 Gaussian distribution𝑦𝑦̇ ha𝑦𝑦vė linear transformation with ( ) Gaussian distribution. Hence, the first derivative of ( ) = ( ) 1 . (12) Weibull random variable has conditional Gaussian 1 𝑐𝑐−1 − 𝛽𝛽𝛺𝛺 𝛺𝛺 𝑐𝑐 distribution. The average value of is: 𝑃𝑃 𝛺𝛺 𝛽𝛽 𝛤𝛤 𝑐𝑐 𝛺𝛺 𝑒𝑒 In this case, the average LCR can be computed from

𝑥𝑥̇ the expression [24]: = ( + ) = 0, (4) 1 𝛼𝛼−1 𝑥𝑥̇̅ 𝛼𝛼𝑥𝑥 𝑦𝑦1𝑦𝑦��̇1� 𝑦𝑦2𝑦𝑦��̇2� = d ( ) = since = = 0 . ∞ 𝑥𝑥 𝑁𝑁 𝑥𝑥 � 𝛺𝛺 𝑛𝑛 �𝛺𝛺𝑃𝑃𝛺𝛺 𝛺𝛺 The𝑦𝑦��̇1� variance𝑦𝑦��̇2� of is: 0 ( ) 1 ( ) = d 𝛼𝛼 = ∞ �/ 1 𝛼𝛼 1 𝑥𝑥̇ 2𝑚𝑚 2 ( ) − 𝛺𝛺 = + , (5) 𝜋𝜋𝑓𝑓 𝑥𝑥 − 𝛺𝛺𝑥𝑥 𝑐𝑐−1 𝛽𝛽 1 2 𝑐𝑐 2 1 2 2 2 2 �0 𝛺𝛺 𝑒𝑒 𝛺𝛺 𝑒𝑒 𝑥𝑥̇ 2 2𝛼𝛼−2 1 𝑦𝑦̇1 2 𝑦𝑦̇2 𝛺𝛺 𝛽𝛽 𝛤𝛤 𝑐𝑐 𝜎𝜎 𝛼𝛼 𝑥𝑥 �𝑦𝑦 𝜎𝜎 𝑦𝑦 𝜎𝜎 � √ 𝜋𝜋 ∞ 1 / ( ) where = d 𝑥𝑥𝛼𝛼 𝛺𝛺 = 2𝑚𝑚 𝛼𝛼 ( ) − − 𝜋𝜋𝑓𝑓 �2 𝑐𝑐−1−1 2 𝛺𝛺 𝛽𝛽 = = 2 = , (6) 𝑐𝑐 𝑥𝑥 �0 𝛺𝛺𝛺𝛺 𝑒𝑒 2 2 2 2 2 2 2 √ 𝜋𝜋 𝛽𝛽 𝛤𝛤 𝑐𝑐 𝑦𝑦̇1 𝑦𝑦̇2 𝑚𝑚 𝑚𝑚 𝜎𝜎 2 𝜎𝜎 𝜋𝜋 𝜎𝜎 𝑓𝑓 𝜋𝜋 𝑓𝑓 𝛺𝛺 = ( ) / 2 , (13) and f is maximal Doppler frequency. ( ) 𝑐𝑐 1 𝛼𝛼 m 𝑚𝑚 𝜋𝜋𝑓𝑓 1 𝛼𝛼� 𝛼𝛼 2−4 𝑥𝑥 If we replace the expression (6) in the expression (5), 𝑐𝑐 2 𝑐𝑐−1 2 √2𝜋𝜋 𝛽𝛽 𝛤𝛤 𝑐𝑐 𝑥𝑥 𝑥𝑥 𝛽𝛽 𝐾𝐾 � � 𝛽𝛽 � the variance is obtained as: where K (x) marks the modified Bessel function of the n 1 second kind [25, eq. (3.471.9)]. = ( + ) = By means of this formula, AFD of wireless 2 2 2 2 2 telecommunication system working over Gamma 𝜎𝜎𝑥𝑥̇ 2 2𝛼𝛼−2 𝜋𝜋 𝑓𝑓𝑚𝑚 𝛺𝛺 𝑦𝑦1 𝑦𝑦2 Ω shadowed Weibull fast fading medium can be calculated. = 𝛼𝛼 𝑥𝑥 xα = . (7) 2 2 Now, the wireless telecommunication system with 1 2 2 𝜋𝜋 𝑓𝑓𝑚𝑚 2 2𝛼𝛼−2 𝑚𝑚 2 2𝛼𝛼−2 dual SC combiner under the action of Gamma shadowing 𝛼𝛼 𝑥𝑥 𝜋𝜋 𝑓𝑓 𝛺𝛺 𝛼𝛼 𝑥𝑥 The joint PDF of Weibull random envelope and its and Weibull multipath fading is considered. The joint first derivative is: PDF of SC combiner output signal envelope and its first derivative is: ( ) = ( ), (8) ( ) = ( ) ( ) + ( ) ( ) = 𝑥𝑥𝑥𝑥̇ 𝑥𝑥̇ 𝑥𝑥̇ 𝑥𝑥 𝑃𝑃 𝑥𝑥𝑥𝑥̇ 𝑃𝑃 � �𝑥𝑥�𝑃𝑃 𝑥𝑥 where ( ) is PDF of Weibull random envelope [2]: 1 1 2 2 2 1 𝑃𝑃𝑧𝑧𝑧𝑧̇ 𝑧𝑧𝑧𝑧̇ 𝑃𝑃𝑧𝑧 𝑧𝑧̇ 𝑧𝑧𝑧𝑧̇ 𝐹𝐹𝑧𝑧 𝑧𝑧 𝑃𝑃𝑧𝑧 𝑧𝑧̇ 𝑧𝑧𝑧𝑧̇ 𝐹𝐹𝑧𝑧 𝑧𝑧 = 2 ( ) ( ), (14) 𝑥𝑥 𝑃𝑃 𝑥𝑥 ( ) ( ) = . (9) 𝑧𝑧1𝑧𝑧̇1 𝑧𝑧2 1 𝛼𝛼 𝑃𝑃 𝑧𝑧𝑧𝑧̇ 𝐹𝐹 𝑧𝑧 𝛼𝛼 𝛼𝛼−1 − 𝛺𝛺𝑥𝑥 with being SC combiner output signal envelope and z1 𝑥𝑥 𝛺𝛺 𝑃𝑃 𝑥𝑥 𝑥𝑥 𝑒𝑒 and z2 are envelopes of input signals in SC combiner, and After substituting (9) in (8), the expression for 𝑧𝑧 joint PDF of Weibull random envelope and the first ( ) ( ) = 1 , 0, (15) derivative of Weibull random envelope becomes: 1 𝛼𝛼 − 𝛽𝛽𝑧𝑧 𝐹𝐹𝑧𝑧2 𝑧𝑧 − 𝑒𝑒 𝑧𝑧 ≥ ( ) ( ) is cumulative distribution function (CDF) of z. ( ) = 2 . (10) 𝑥𝑥̇ 1 𝛼𝛼 1 − 2𝜎𝜎𝑥𝑥̇ 2 𝛼𝛼 𝛼𝛼−1 − 𝛺𝛺𝑥𝑥 𝑥𝑥𝑥𝑥̇ 𝑃𝑃 𝑥𝑥𝑥𝑥̇ √2𝜋𝜋𝜎𝜎𝑥𝑥̇ 𝑒𝑒 𝛺𝛺 𝑥𝑥 𝑒𝑒 Tehnički vjesnik 23, 6(2016), 1579-1584 1581 Level crossing rate of SC receiver over gamma shadowed Weibull multipath fading channel S. Suljovic et al.

After substitutions (10) and (15) in (14), the formula LCR has lower values for bigger values of Weibull for the joint PDF of selection combiner output signal multipath fading severity parameter and then system envelope and the first derivation becomes: performance is better. In Fig. 2, the average LCR is plotted versus Gamma shadowing severity parameter c for a few magnitudes of ( ) β SC combiner output signal envelope, Weibull multipath ( ) = ̇ 2 1 .(16) 𝑧𝑧 1 1 𝛼𝛼 − 2 𝛼𝛼 fading severity parameter α, and average power of 1 2𝜎𝜎𝑥𝑥̇ 𝛼𝛼 𝛼𝛼−1 −𝛺𝛺𝑧𝑧 − 𝑧𝑧 𝑧𝑧𝑧𝑧̇ 𝑥𝑥̇ Gamma random variable b. 𝑃𝑃 𝑧𝑧𝑧𝑧̇ 𝑧𝑧 √2𝜋𝜋𝜎𝜎 𝑒𝑒 𝛺𝛺 𝑧𝑧 𝑒𝑒 � − 𝑒𝑒 � Average LCR increases as Gamma shadowing The LCR of SC combiner output signal is: severity parameter grows. For bigger dimensions of

∞ Weibull multipath severity parameter α, average level = d ( ) = crossing rate has lower sizes. Also, average LCR has lower values for greater values of average power of 𝑧𝑧2 ( 𝑧𝑧𝑧𝑧̇ ) ( ) Gamma random variable b and SC combiner output signal 𝑁𝑁= �0 𝑧𝑧̇ 𝑧𝑧̇𝑃𝑃 𝑧𝑧𝑧𝑧1̇ 1 1 𝛼𝛼 𝛼𝛼 − 𝑧𝑧 envelope z. 𝛼𝛼 𝛼𝛼−1 − 𝛺𝛺𝑧𝑧 𝛽𝛽 ∞ 𝑧𝑧 𝑒𝑒1 � − 𝑒𝑒 � ∙ 𝛺𝛺d 𝑧𝑧̇ 2 = 1 2 3 2 −�2𝜎𝜎 2� 𝑥𝑥̇ 0.09 a=1.5 b=1.5 0.09c=1.5 / ∙ �2 𝑧𝑧̇𝑧𝑧̇ 𝑒𝑒 a=2 b=2 c=2 0 ( 𝑥𝑥̇ ) ( ) 0.08 0.08 = √ 𝜋𝜋𝜎𝜎 1 = a=3 b=2 c=2 1 𝛼𝛼 1 𝛼𝛼 1 2 − 𝑧𝑧 2 0.07 0.07 𝛼𝛼 𝛼𝛼−1 − 𝛺𝛺𝑧𝑧 𝛽𝛽 𝜋𝜋𝑓𝑓𝑚𝑚𝛺𝛺 ( ) 𝛼𝛼� −1 = 𝑧𝑧 𝑒𝑒 ( �) 1− 𝑒𝑒 � . 2 (17) 0.06 0.06 𝛺𝛺 𝛼𝛼�/ 1 1 𝛼𝛼 √ 𝜋𝜋𝑧𝑧 𝛼𝛼 𝑚𝑚 2 𝛼𝛼 2𝜋𝜋𝑓𝑓 𝑧𝑧 − 𝛺𝛺𝑧𝑧 − 𝛽𝛽𝑧𝑧 0.05 0.05 1 2 LCR √2𝜋𝜋 𝛺𝛺 𝑒𝑒 � − 𝑒𝑒 � 0.04 0.04 The power of Weibull signal envelope is Gamma distributed. The LCR of SC combiner output signal 0.03 0.03 envelope in Gamma shadowed Weibull short term fading 0.02 0.02 environment is: 0.01 0.01 0.00 0.00 ∞ 1 2 3 z = d / ( ) = Figure 1 LCR versus signal envelope 𝑧𝑧 ∞ 2 𝑧𝑧 𝛺𝛺 𝛺𝛺 𝑁𝑁 �0 𝛺𝛺 𝑁𝑁 𝑃𝑃 ( 𝛺𝛺 ) = 𝛼𝛼 1 2 3 �2 1 𝛼𝛼 0.095 0.095 2 𝑚𝑚 − 𝑧𝑧 𝜋𝜋𝑓𝑓 𝑧𝑧 𝛺𝛺 0.090 0.090 1 1 � ( ) �2 𝑒𝑒 ∙ ( ) 0.085 0.085 10 = 0.080 0.080 𝜋𝜋1 1 √ 𝛺𝛺𝛼𝛼 (0) 0.075 0.075 − 𝑧𝑧 − 𝛽𝛽𝛺𝛺 𝛺𝛺 𝑐𝑐−1 0.070 0.070 ∞ 𝑐𝑐 z=1.5 b=1.5 a=1.5 ∙ � − 𝑒𝑒2 � 𝛺𝛺 𝑒𝑒 ( ) ( ) ( ) 0.065 z=2 b=2 a=20.065 0.060 0.060 = 𝛽𝛽d 𝛤𝛤 1 = (0) 𝛼𝛼 1 1 𝛼𝛼 2 𝛼𝛼 0.055 z=2.5 b=2 a=30.055 𝑚𝑚 𝑐𝑐−1− − 𝑧𝑧 − 𝑧𝑧 − 𝛽𝛽𝛺𝛺 𝑓𝑓 √ 𝜋𝜋 2 2 𝛺𝛺 𝛺𝛺 0.050 0.050 𝑐𝑐 ∞ LCR 2 𝑧𝑧 � 𝛺𝛺 𝛺𝛺 �𝑒𝑒 ( − )𝑒𝑒 ( � 𝑒𝑒 ) 0.045 0.045 0 0.040 0.040 =𝛽𝛽 𝛤𝛤 d 𝛼𝛼 𝛼𝛼 = 𝛼𝛼 1 𝑧𝑧 𝛺𝛺 2𝑧𝑧 𝛺𝛺 0.035 0.035 (0) − − − − 𝑓𝑓𝑚𝑚√ 𝜋𝜋 2 𝑐𝑐−1−2 𝛺𝛺 𝛽𝛽 𝛺𝛺 𝛽𝛽 0.030 0.030 𝑐𝑐 0.025 0.025 2 𝑧𝑧 �0 𝛺𝛺 𝛺𝛺 ∙ �𝑒𝑒 − 𝑒𝑒 � = 𝛽𝛽 𝛤𝛤 ( ) 2 2 (2 ) 0.020 0.020 (0) 𝛼𝛼 𝑐𝑐 1 / 𝑐𝑐 1 0.015 0.015 𝑓𝑓𝑚𝑚√ 𝜋𝜋 2 𝛼𝛼 2−4 𝛼𝛼 2−4 0.010 0.010 𝑐𝑐 𝑐𝑐−1 2 𝛼𝛼 0.005 0.005 𝑧𝑧 � 𝛽𝛽𝑧𝑧 𝐾𝐾 � � � �� 𝛽𝛽𝑧𝑧 ∙ 1 2 3 2 / 𝛽𝛽 𝛽𝛽 𝛤𝛤 2 2 = ( ) c / (0) 𝑐𝑐 1 𝛼𝛼 𝑚𝑚 𝛼𝛼 2 𝛼𝛼 2−4 Figure 2 LCR versus shadowing severity 𝑐𝑐−1 2 𝑓𝑓 √ 𝜋𝜋 𝑧𝑧α 𝑐𝑐 α ∙ 𝐾𝐾 � � �𝛽𝛽� 𝑧𝑧 𝛽𝛽𝑧𝑧 ∙ / 2 𝛽𝛽 K𝛤𝛤 / 2 . (18) β 𝑐𝑐 1 β The system properties are more favorable for smaller 𝑧𝑧 2−4 2𝑧𝑧 𝑐𝑐−1 2 c−1 2 values of average LCR. The obtained results are drawn to ∙ �𝐾𝐾 � � � − 𝑧𝑧 � � �� explain the impact of Weibull multipath fading severity 4 Numerical results parameter and Gamma slow fading severity parameter on average LCR. The average LCR of SC combiner output signal The best performance is achieved with a small envelope is given depending on the SC combiner output average LCR, namely with bigger SC combiner output signal envelope for a number of values of Weibull signal envelope z, Weibull multipath fading severity multipath fading severity parameter α (a in figures), parameter α, average power of Gamma random variable b Gamma slow fading severity parameter c and average and with declining of Gamma shadowing severity power of Gamma random variable b in Fig. 1. parameter c. The average LCR increasing for smaller values of SC combiner output signal envelope z, reaches the maximum, 5 Conclusion then decreases.The impact of SC combiner output signal envelope on average LCR is greater for higher values of The wireless telecommunication system with SC Weibull multipath fading severity parameters. Average combiner with two branches, working over channel under

1582 Technical Gazette 23, 6(2016), 1579-1584 S. Suljovic i dr. Srednji broj osnih presjeka kod SC prijemnika u kanalu s Weibull-ovim fedingom i Gamma sjenkom presence of slow and fast fading is analyzed. The received Vehicular Technology. 56, 1(2007), pp. 27-34. DOI: signal suffers Gamma shadowing and Weibull fading 10.1109/TVT.2006.883753 leading in system performance impairment. Weibull small [9] Kostic, I. Analytical approach to performance analysis for scale fading triggers off variation of the signal envelope channel subject to shadowing and fading. // IEEE Proc. Comm. 152, (2005), pp. 821-827. and Gamma large scale fading induces changes of the [10] Sekulovic, N. M.; Stefanovic, M. C. Performance analysis signal envelope power. SC combiner is used for reducing of system with micro- and macrodiversity reception in the long and short term fading influences on system corelated gamma shadowed Rician fading channels. // properties. Wireless Personal Commun. 65, 1(2012), pp. 143-156. DOI: In this paper, the useful formula for the average LCR 10.1007/s11277-011-0232-8 of SC combiner output signal envelope in the closed form [11] Hansen, F.; Meno, F. I. Mobile Fading-Rayleigh and is determined. This expression was found as an average Lognormal Superimposed. // IEEE Transactions on value of the first derivative of the SC combiner output Vehicular Technology. VT-26, 4(1977), pp. 332-335. DOI: signal envelope. The joint PDF of SC combiner output 10.1109/T-VT.1977.23703 [12] Gomez-Deniz, E.; Gomez-Deniz, L. A generalisation of the signal envelope and its first derivative is also established. Rayleigh distribution with applications in wireless fading The average LCR grows for small values of SC channels. // Wireless Communications and Mobile combiner output signal envelope and declines for bigger Computing, Published online 8 February 2011 in Wiley values of SC combiner output signal envelope. For lower Online Library. 13, 1(2013), pp. 85-94. DOI: values of Weibull multipath fading severity parameter and 10.1002/wcm.1097 lower values of Gamma shadowing severity parameter SC [13] Prasad, R.; Kegel, A. Effects of Rician faded and log- receiver output has a larger impact on the average LCR. normal shadowed signals on spectrum efficiency in Therefore, for better system performance it is good to microcellular radio. // IEEE Transactions on Vehicular choose higher values for: SC combiner output signal Technology. 42, 3(1993), pp. 274-281. DOI: 10.1109/25.231878 envelope z,Weibull multipath fading severity parameter α, [14] Tjhung,T. T.; Choy, C. C.; Dong, X. Outage probability for and average power of Gamma random variable b and lognormal-shadowed Rician channels. // IEEE Transactions lower for Gamma shadowing severity parameter c. on Vehicular Technology. 46, 2(1997), pp. 400-407. DOI: The obtained results areuseful in performance 10.1109/25.580779 analyzing and production of wireless telecommunication [15] Sekulović, N. M.; Stefanović, M. Č. Performance Analysis systems with SC combiners which work per channels of System with Micro- and Macrodiversity Reception in under the effects of Gamma shadowing and Weibull Correlated Gamma Shadowed Rician Fading Channels. // fading. By selecting optimal parameters, the best Wireless Personal Communications. 65, 1(2012), pp. 143- performance could be achieved and the reliable systems 156. DOI: 10.1007/s11277-011-0232-8 [16] Stefanovic, D.; Panic, S. R.; Spalevic, P. Second-order obtained. statistics of SC macrodiversity system operating over Gamma shadowed Nakagami-m fading channels. // AEU - Acknowledgements International Journal of Electronics and Communications. 65, 5(2011), pp. 413-418. DOI: 10.1016/j.aeue.2010.05.001 This article is realised in the framework of projects [17] Bandjur, M. V.; Bandjur, Dj. V. Performance Analysis of TR-33035 and III-44006 of the Ministry of Education, SSC Diversity Receiver over Correlated Hoyt Fading Science and Technological Development of Serbia. Channels. // Radioengineering. 21, 1(2012), pp. 110-114. [18] Baid, A.; Fadnavis, H.; Sahu, P. R. Performance of a 6 References Predetection EGC Receiver in Hoyt Fading Channels for Arbitrary Number of Branches. // IEEE Communication

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Tehnički vjesnik 23, 6(2016), 1579-1584 1583 Level crossing rate of SC receiver over gamma shadowed Weibull multipath fading channel S. Suljovic et al.

[24] Gradshteyn, I.; Ryzhik, I. Tables of Integrals, Series, and products. Academic Press, New York 1994. [25] Modified Bessel Function of the Second Kind, http://mathworld.wolfram.com/ModifiedBesselFunctionoft heSecondKind.html

Authors’ addresses

Suad Suljović Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, 18000 Niš, Serbia E-mail: [email protected]

Dragana Krstić Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, 18000 Niš, Serbia Fax: +38118588399 E-mail: [email protected]

Srdjan Maričić Faculty of Industrial Management, University Union Belgrade, Kosančićev venac 2, 11000 Belgrade, Serbia E-mail: [email protected]

Srboljub Zdravković Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, 18000 Niš, Serbia E-mail: [email protected]

Vladeta Milenković Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, 18000 Niš, Serbia E-mail: [email protected]

Mihajlo Stefanović Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, 18000 Niš, Serbia E-mail: [email protected]

1584 Technical Gazette 23, 6(2016), 1579-1584