IJRIT International Journal of Research in Information Technology, Volume 3, Issue 1, January 2014, Pg. 41-47

International Journal of Research in Information Technology (IJRIT) www.ijrit.com

ISSN 2001-5569

Outage and ASER Characterization of κ - µ Fading Channels. Aheibam Dinamani Singh1, N. Loyalakpa Meitei2 1

2

Assistant Professor, NERIST, Department of ECE, Nirjuli, Arunachal Pradesh-791 109, India. Email: [email protected]

Assistant Professor, NERIST, Department of ECE, Nirjuli, Arunachal Pradesh-791 109, India. Email: [email protected]

Abstract The performance evaluation of communication systems wireless communication is important. However, the physical performance measurement of the systems is costly and sometimes unfeasible. Hence, performance evaluation by analytical method is popular. The outage probability and average symbol error rate (ASER) are important performance measure parameters. In this paper, the performance of a single channel system with different digital modulation schemes are analyzed over κ - µ fading channels. Expressions of outage probability and ASERs have been derived using the probability distribution function based approach. The performance expressions derived are numerically evaluated and verified with Monte Carlo simulation.

Keywords: ASER, κ - µ fading, M-ary modulation, Outage probability, PDF.

1. Introduction The occurrence of fading in the wireless channels degrades the performance of wireless communication systems. One of the major challenges that a system designer has to face is to eliminate or diminish this undesirable effect. Different modulation schemes have different amount robustness to the fading characteristics of the channels. The small scale fading encountered in mobile radio channels with and without line-of-sight components is suitably modeled with the help of κ - µ distribution [1, 2, 3]. The benefit of this model is because of a nonhomogenous physical environment assumption. Such assumption makes it closer to the practical scenario and fits well to the experimental data. Moreover, other known fading models such as Rayleigh, Rice, Nakagami-m etc. can also be realized as particular cases of this model. In [4], the moment generating functions (MGFs) of the generalized κ - µ and η-µ distribution are expressed in terms of elementary functions. The average bit error rate (ABER) formulas for various modulation formats are derived using the MGF approach. The outage performance of L-branch maximal-ratio combiner (MRC) for generalized κ - µ fading is presented in paper [5]. An exact closed-form expression for phaseenvelope joint distribution of the κ - µ fading environment is derived in [6]. The outage and symbol error probability (SEP) performance of dual-branch MRC in presence of generalized κ - µ fading was treated in the paper. The performance of a selection combiner (SC) receiver over κ - µ and η-µ fading channels for arbitrary number of fading branches has been analyzed in [7]. In the paper, the PDF based approach has been used to obtain a set of new mathematical expressions for moments of SC output SNR and ABER performance for binary, coherent -phase shift keying (CPSK), frequency shift keying (CFSK), differential

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IJRIT International Journal of Research in Information Technology, Volume 3, Issue 1, January 2014, Pg. 41-47

coherent phase shift keying (DPSK) and non-coherent FSK (NCFSK) modulation schemes. Comparisons have been performed between κ - µ and Rayleigh fading channels was carried out in [8]. A generalized Laguerre polynomial expansion for the PDF and the cumulative distribution function (CDF) of the sum of independent non-identically distributed squared κ - µ random variables is proposed in [9]. A natural generalization of κ - µ fading channel in which the LOS component is subjected to shadowing is investigated in [10]. However, to the best of the authors’ knowledge the outage and ASER analysis of a single channel system over κ - µ using the PDF approach have not been reported. This motivates to evaluate these parameters. In this paper, the probability distribution function (PDF) of the output SNR of a single channel receiver over κ - µ fading channels is derived. The PDF expression is used to derive the expressions of the various performance parameters such as outage probability and ASER for different coherent and non-coherent digital modulation schemes. These expressions are numerically evaluated and validated either by Monte Carlo simulation or available published results. The rest of this paper is organized as follows. In Section 2, the channel and system considered for analysis has been discussed. In Section 3, performance parameters of a single channel receiver have been obtained over κ - µ fading channels. In Section 4, numerical results and discussion are presented. Finally, the paper is concluded in Section 5.

2. Channels and Systems The channel has been assumed to be slow, frequency nonselective, with κ – µ fading statistics. The complex low pass equivalent of the received signal over one symbol duration Ts can be expressed as

r / ( t ) = re jϕ s ( t ) + n ( t ) ,

(1)

where s(t) is the transmitted symbol with energy Es and n(t) is the complex Gaussian noise having zero mean and two sided power spectral density 2N0. Random variable φ represents the phase and “r” is the κ µ distributed The PDF of the received SNR at the receiving antenna for κ - µ fading channel is given in [7] as

fγ ( γ ) =

µ (1 + κ )

µ +1

γ

2

µ −1

κ

2

µ −1 − µ (1+κ )γ γ 2

e

µ +1

γ

2

e

µκ

 κ (1 + κ ) γ I µ −1  2 µ  γ 

 ,  

(2)

where parameter µ is defined as the ratio between the total power of the dominant components and the total power of the scattered waves. Parameter κ denotes the multipath clusters and Iv(·) is the Bessel function of the first kind and vth order.

3. Performance of Single channel Receiver In this section, the expression of outage probability and ASER for both the coherent and non-coherent Mary modulations schemes over κ - µ fading channel are derived.

3.1 Outage probability Outage probability Pout is the probability that the instantaneous error probability exceeds a specific value [11]. In other words, it is the probability that the output SNR (γ) drops below a certain pre-defined threshold SNR (γth). Outage probability provides a fundamental performance measure of the grade of service for cellular mobile radio systems [12]. The expression of the outage probability is derived from the expression of the output SNR (2) by integrating over the limit 0 to γth, with respect to γ. The resulting expression is µ +1

Pout ( γ th ) =

µ (1 + κ ) µ −1

κ

2

2

µ +1

γ

2

e

γ th

∫γ

µκ 0

µ −1 − µ (1+κ )γ γ 2

e

 κ (1 + κ ) γ I µ −1  2 µ  γ 

  dγ ,  

(3)

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IJRIT International Journal of Research in Information Technology, Volume 3, Issue 1, January 2014, Pg. 41-47

The integration in the expression of Pout can be solved [13, (3.351.1)], and the expression is expressed as,

1 Pout ( γ th ) = µκ e

µ 2 n + µκ n  1 + κ  ∑   n = 0 n !Γ ( n + µ )  γ  ∞

n+µ

 µ (1 + κ )    γ  

−( n + µ )

 µ (1 + κ )  g  n + µ, γ th  , γ  

(4)

where Γ(·) is gamma function and g(·, ·) is lower incomplete gamma function. The simplified expression of Pout can be expressed as

Pout ( γ th ) =

1 e µκ

∞



µ nκ n

∑ n !Γ ( n + µ ) g  n + µ , 

n=0

µ (1 + κ )  γ th  . γ 

(5)

3.2 Average Symbol Error Rate The ASER of a digital communication system for various M-ary modulations can be obtained by averaging the conditional symbol error rate (SER) corresponding to the modulation over the PDF of the receiver output SNR [14]. Mathematically, ASER can be given as ∞

Pe γ = ∫ pe ( ε γ ) fγ ( γ ) d γ ,

()

(6)

0

where pe (ε |γ) is the conditional SER corresponding to the modulation scheme used. The conditional SER for different digital M-ary modulation schemes are available in literature. The ASER expressions for coherent and non-coherent modulation schemes are derived below.

3.2.1 Coherent Modulations For coherent modulations, the expression for the conditional SER can be given as

pe ( ε γ ) = aQ

(

)

bγ ,

(7)

where parameter a and b are chosen as per the modulation scheme used, and tabulated in Table 1. Table 1: Values of a and b for some coherent modulation. a  M −1  b 4   .5 1 

M



1 2

BFSK -

BPSK

-

2sin 2 (π M )

-

MPSK

-

3 M −1

-

-

Rect.QAM

The Q function can be expressed as Q ( bx ) =

1 b  Γ  , x  [11]. Putting (2) and (7) in (6), the Pe,ch 2 π 2 2  1

expression is obtained as

a Pe.ch ( γ ) = 2π e µκ

µ 2 n + µκ n  1 + κ  ∑   n = 0 n !Γ ( n + µ )!  γ  ∞

n+µ ∞

∫γ 0

n + µ −1

e

−

µ (1+ κ ) γ γ

1 b  Γ  , x  dγ , 2 2 

(8)

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IJRIT International Journal of Research in Information Technology, Volume 3, Issue 1, January 2014, Pg. 41-47

Solving the integral using [13, 6.455.1], an expression for ASER can be obtained as

a b Pe.ch ( γ ) = 2 2π e µκ

∞

∑ ( 2γ ) n =0

η

1+ κ  ×   γ 

2

Γ (η + 0.5 ) µ 2 n + µκ n

η + 0.5

η + 0.5

n!η !( γ b + 2 µ (1 + κ ) )

 2 µ (1 + κ )  F1 1,η + 0.5;η + 1;   γ b + 2 µ (1 + κ )  

(9)

where 2F1(a;b;c; z) is hypergeometric function and η = n+µ.

Figure 1: Outage Probability for κ−µ Fading Channels with γth = 2dB.

3.2.2. Non-coherent Modulations For non-coherent modulations, the conditional SER is given as

pe ( ε γ ) = a e− bγ

(10)

The value of a and b depends on modulation format used, which is tabulated in Table 2 for convenience. By putting (2) and (10) together in (6), the expression for Pe,nch is obtained as,

a Pe.nch ( γ ) = µκ e

µ 2 n + µκ n  1 + κ  ∑   n = 0 n !Γ ( n + µ )!  γ  ∞

n+ µ ∞

∫γ

n + µ −1

e

 µ (1+ κ )  − + bγ  γ 

dγ .

(11)

0

The integral can be solved using [13, (3.351.3 )], and after simplification the expression for ASER can be obtained as

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IJRIT International Journal of Research in Information Technology, Volume 3, Issue 1, January 2014, Pg. 41-47

aγ . Pe, nch ( γ ) = µκ e

  µ n −1κ n γ b ( n + µ ) 1 + ∑ n = 0 n !(1 + κ )  µ (1 + κ )  ∞

− ( n + µ +1)

(12)

Figure 2: ASER of Coherent Digital Modulations over κ−µ Fading Channels.

4. Numerical Results and Discussion Numerical analyses of the expressions of outage probability and ASERs obtained in Section 3 have been carried out. The results are plotted for different digital modulations, different values of fading parameters κ and µ for the purpose of illustration.

Table 2: Values of a and b for some non-coherent modulation. b a 0.5 1 0.5 BFSK DBPSK

( M − 1)

2

MFSK

-

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IJRIT International Journal of Research in Information Technology, Volume 3, Issue 1, January 2014, Pg. 41-47

Figure 3: ASER of Digital Modulations. In Fig. 1, outage probability vs. average received SNR per branch ( γ ) have been plotted. The plot is made for different values of κ and µ. It can be seen from the figure that for the same value of κ the outage performance improves with the increase in the value of µ. This is because parameters µ is the real extension of number of clusters [3]. Also, the outage performance improves with the increase in the value of κ. The parameter κ indicates the power of the dominant line of sight component and hence higher value of κ indicated better channels. In Fig. 3, the ASER vs average received SNR per branch ( γ ) have been plotted for coherent system. The plot is for a constant value of κ = 1 and µ = 0:5. It is observed that the ABER performance of BPSK is better than that of BFSK in the fading channels. The information is stored in the phase variations of the transmitted signal in the case of BPSK, while in BFSK, the information is stored in the frequency variations. The frequency of signal can be easily affected by noise but not the phase hence; BPSK is performing better than BFSK. At the same time, the distance between the message points is greater in BPSK. This gives lesser probability of taking a wrong decision in favour of one symbol in presence of fading. But, the ASER performance degrades as the order of modulation increases, since the messages points are becoming more closure to each other. As the order of modulation increases its bit rate also increases. Therefore, a fade of a particular duration will cause more number of bits in error for higher order modulations. The ASER performance of QAM is worse than that of QPSK. It is so because in QAM message is store in the amplitude of the carrier which can be deeply affected by fading. In Fig. 12, the plot of ASER vs average received SNR per branch ( γ ) for non-coherent system and coherent BFSK for comparison. The ABER performance of coherent BFSK is better that that of noncoherent BFSK. At an average SNR of 12dB the ABER of 1.9×10−2 and 1.06×10−1 are observed for coherent and non-coherent BFSK respectively. This is an expected outcome because of the fact that

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IJRIT International Journal of Research in Information Technology, Volume 3, Issue 1, January 2014, Pg. 41-47

coherent detection is ideal one which needs perfect phase and frequency synchronization. However, in many applications stringent phase synchronization is not possible, that is why the ASER plots of other modulation schemes are also presented in the figure. The observations for higher order modulation schemes are similar as above. Computer simulated results have been included in the figures. The results are in close agreement with the numerical results.

5. Conclusions In this paper, the expressions for the outage probability and ASER of a single channel system over κ - µ fading channel have been derived. The expression for the outage probability is derived using the characteristic function, whereas the expressions for ASER for both the coherent and non-coherent modulation schemes are derived by using the PDF of output SNR of the system and the conditional error probability of the modulation schemes. The results are verified with Monte Carlo simulation.

References [1] M. D. Yacoub, “The η - µ distribution: a general fading distribution, IEEE Boston Fall VTC Conf. 2000, Boston, USA, Sep. 2000. [2] Michel Daoud Yacoub, “The κ-µ distribution: A General Fading distribution.” IEEE Atlantic City Fall VTC Conf. 2001, Atlantic City, USA, Oct. 2001. [3] M. D. Yacoub, “The κ-µ distribution and the η - µ distribution, IEEE Ant. Prop. Mag., vol. 49, no. 1, pp. 68-81, Feb. 2007. [4] N.Y. Ermolova , “ Moment Generating Function s of the generalized η-µ and κ-µ distributions and their applications to performance evaluations of communication systems”. IEEE communications letters , vol.12, no. 7, pp: 502 - 504, July 2008. [5] Mirza Milisic , Mirza Hamza, Mesud Hadzialic, “Outage Performance of L Branch Maximal-Ratio Combiner for Generalized κ-µ Fading”. 50th International Symposium ELMAR-2008, 10-12 September 2008, Zadar, Croatia [6] Ugo Silva Dias and Michel Daoud Yacoub, “The κ-µ Phase-Envelope Joint Distribution”. IEEE Trans. on Commn., VOL. 58, NO. 1, pp: 40 - 45, JANUARY 2010. [7] Rupban Subadar, T.Siva Bhasker Reddy, P.R.Sahu, “Performance of an L-SC Receiver over κ-µ and ηµ Fading channels”. in Proc. of IEEE ICC 2010 23-27 May 2010, Cape Town, South Africa. [8] Diogo Sanders, Fabio von Glehn, and Ugo Silva Dias, “Spectrum sensing over κ-µ fading channels”. in XXIX SIMP OSIO BRASILEIRO DE TELECOMUNICAC, OES - SBrT 2011, DE 2 A 5 DE OUTUBRO DE 2011, CURITIBA, PR. [9] Kostas P. Peppas, “ Sum of Nonidentical Squared κ-µ Variates and Applications in the Performance Analysis of Diversity Receivers”. IEEE Trans. on Veh. Tech. VOL. 61, NO. 1, Jan. 2012. [10] Jose F.Paris, “Statistical Characterization of κ-µ shadowed fading”. arXiv: 1304.7435v1[cs.IT] 28 Apr 2013. [11] V. A. Aalo, “Performance of maximal-ratio diversity systems in a correlated Nakagami-fading environment,” IEEE Trans. on Commun., vol. 43, No. 8, pp. 2360-2369, Aug. 1995. [12] Q. T. Zhang, “Outage probability in cellular mobile radio due to Nakagami signal and interferers with arbitrary parameters.” IEEE Trans. on Veh. Technol., vol.45, no.2, pp. 364-372, May. 1996. [13] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., San Diego, CA: Academic, 2000. [14] M. K. Simon and M. -S. Alouini, Digital Communications Over Fading Channels, 2nd ed. New York: Wiley, 2005.

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