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IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 1, NO. 4, AUGUST 2012
Performance of L-Branch MRC Receiver in -µ and -µ Fading Channels for QAM Signals Dharmendra Dixit and P. R. Sahu, Member, IEEE
Abstract—Error performance of a L-branch maximal ratio combining (MRC) receiver in -µ and -µ fading channels is analyzed. A simple and highly accurate approximation to the average symbol error rate (ASER) expression of quadrature amplitude modulation (QAM) scheme is derived. Steps to extend the approach to analyze ASER performance of coherent, differentially encoded quadri-phase shift-keying (DE-QPSK) and /4-QPSK schemes are also stated. Numerically evaluated results are plotted and compared with Monte Carlo simulation to verify the accuracy of the derivation. Index Terms—-µ, -µ, MRC, ASER, QAM.
I. I NTRODUCTION HE -µ and -µ fading distributions are generalized distributions which can be used to characterize fading channels for a more comprehensible range of values of measurable physical parameters compared to the other known fading distributions. Further, these models fit to experimental data, closer than the fit offered by popular fading distributions such as Rice, Hoyt, and Nakagami-m etc., even in tail regions [1]. A number of research works on the error performance of communication systems operating over -µ and -µ fading models have been presented in [2]-[7]. In [2], average symbol error rate (ASER) expressions are obtained using the moment generating function (MGF) based approach. In [3], MGFs of generalized -µ and -µ distributions are presented and ASER expressions are given either in the form of elementary functions or in the form of finite-limit integrals of elementary functions. In [4], closed form expression for ASER for rectangular quadrature amplitude modulation (QAM) is obtained in the form of Appell’s (F1 (.) and 1 (.)) and Lauricella’s (3) (3) (FD (.) and 1 (.)) hypergeometric functions. In [5], exact expressions for ASER of various digital modulation schemes with maximal ratio combiner (MRC) over L independent, not necessarily identically distributed -µ fading channels have been presented. An ASER expression for the MRC receiver over − µ fading channels has been presented in [6] using an approximation for Gaussian Q-function. In [7], ASER of L-branch MRC in generalized -µ fading channels is analyzed for non-coherent modulation schemes. In this letter, we present ASER of a L-branch MRC receiver for QAM modulated signal over -µ and -µ fading channels.
T
Manuscript received March 31, 2012. The associate editor coordinating the review of this letter and approving it for publication was G. Colavolpe. D. Dixit is with the Department of Electronics and Communication Engineering, Jaypee University of Engineering and Technology, Raghogarh, Guna, Madhya Pradesh, India (e-mail:
[email protected]). P. R. Sahu is with the Department of Electronics and Electrical Engineering, Indian Institute of Technology Guwahati, Assam, India (e-mail:
[email protected]). Digital Object Identifier 10.1109/WCL.2012.042512.120240
We employ an useful exponential bound for the error function er f c(·) given in [8] to evaluate the average of one Gaussian Qfunction or the product of two Gaussian Q-functions over -µ and -µ fading distributions. The approximate solution is quite simple and does not put any restriction on fading parameters. Therefore, it can be used to analyze ASER performance under different fading severity conditions. The bandwidth efficient rectangular QAM modulation is a generic modulation technique since it includes square QAM, binary phase-shift keying (BPSK), orthogonal binary frequency-shift keying, quadrature phase-shift keying and multilevel amplitude shift-keying modulations as special cases [9]. The approach presented here can also be utilized to analyze ASER performance of coherent, differentially encoded quadri-phase shift-keying (DE-QPSK) and /4-QPSK schemes. The letter is organized as follows. Section II describes channel model and the error performance analysis is presented in Section III. In Section IV, numerical results and discussion are given. The letter is concluded in Section V. II. C HANNEL M ODEL The channel is assumed to be slow and flat fading with -µ and -µ statistics. For a transmitted signal s(t) with symbol energy Es , the complex low pass equivalent of the received signal at the lth (l = 1, 2, . . . , L) path over a symbol duration Ts second can be expressed as rl (t) = l e jl s(t) + nl (t), where nl (t) is the complex additive white Gaussian noise having one sided power spectral density N0 , random variable (RV) l is the instantaneous phase and RV l is the fading envelope which is either − µ or − µ distributed as discussed below. A. -µ Fading Distribution The probability density function (PDF) of -µ distributed RV is given by [1] √ 1 2µ 2 4 µµ+ 2 hµ l − 2µh 2µH 2 l l e I (1) p−µ (l ) = µ− 21 1 µ+ 1 l l (µ)H µ− 2 l 2 where l = E[2l ], E[·] is the expectation operator, (·) is the gamma function, and h and H are functions of the parameter . The parameter is defined under two Formats: In Format 1, it is assumed that the in-phase and quadrature phase components of the fading signal within each cluster are independent of each other and have different average powers. The parameter ∈ (0, ) is the ratio of these powers, and h = (2 + −1 + )/4 and H = (−1 − )/4. In Format 2, it is assumed that the in-phase and quadrature phase components within each cluster are correlated and have identical powers. The parameter
c 2012 IEEE 2162-2337/12$31.00
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∈ (−1, 1) is the correlation coefficient between these components with h = 1/(1 − 2 ) and H = /(1 − 2 ). In both the formats, the parameter µ denotes the number of multipath clusters. This fading model includes Hoyt ( = q2 , µ = 12 ), Nakagami-m ( = 1, µ = m/2), Rayleigh and one sided Gaussian distribution as special cases. The -µ distribution is better suited for non-line-of-sight signal propagation.
(5)
The PDF of -µ distributed RV is given by [1] p−µ (l ) =
2µ(1 + ) µ−1 2
(2)
where > 0 and µ > 0 are the parameters of the distribution and I (·) is the modified Bessel function of the first kind and th order. The parameter is the ratio of the total power due to dominant components to the total power due to scattered waves and µ is the number of multipath clusters. This fading model includes Rice (µ = 1 and = K), Nakagami-m ( → 0 and µ = m), Rayleigh (µ = 1 and → 0) and one sided Gaussian distribution (µ = 0.5 and → 0) fading models as special cases. This distribution is better suited for line-of-sight (LOS) signal propagation.
For equally-likely transmitted symbols, the output SNR, t of a L-branch MRC receiver is given as t = Ll=1 l , where l = NEos 2l is the instantaneous received SNR at the lth input branch [10]. For the performance analysis purpose we derive expressions for the PDF of t as discussed in sections below. A. -µ Fading Channels 1) PDF of MRC Output SNR: In [1] it is shown that the sum of L independent identically distributed (-µ)-square variates is (-Lµ)-square distributed. Hence, the PDF of t can be obtained by first obtaining the PDF of (-µ)-square variate from (1), scaling it by the factor Es /N0 and then substituting Lµ in place of µ in the resulting expression. Thus, the final expression for the PDF of t can be given as √ t 2 hLµ Lµ Lµ+ 12 t Lµ− 12 −( 2Lµh t ) p−µ (t ) = e (Lµ) t H 2LµHt × ILµ− 1 , (3) 2 t where t = E [t ]. 2) Average Symbol Error Rate for Rectangular QAM: The ASER can be obtained by averaging the conditional symbol error rate for additive white Gaussian channels over the PDF of the output SNR. Mathematically, ASER P(e), for any modulation scheme can be given as 0
P(e|t )p (t )dt ,
(4)
Q
b = a and = dQ /dI with dI and dQ being the in-phase and quadrature phase decision distance, respectively. We know that Q-function is related to the complementary error function erfc(x) as Q(x) = 12 erfc √x2 . A simple but tight exponential upper bound for erfc(x) is presented as [8, eq. 14] 2 1 2 1 erfc(x) e−x + e−4x /3 , 6 2
x > 0.5.
(6)
Thus, using (3), (5) and (6) in (4) and solving the integral, an expression for the ASER can be obtained as shown in (7) at the top of the next page, where 1 () = 0 e−t p−µ (t )dt . To obtain a closed form expression for (7) it is required to solve 1 (). Applying [12, (29.3.60)] and simplifying, a solution for 1 () can be given as ⎤Lµ 2 h( 2Lµ t ) 1 () = ⎣ 2 2LµH 2 ⎦ . + 2Lµh − ⎡
t
III. P ERFORMANCE A NALYSIS
P(e) =
√ √ 2pQ(a t ) + 2qQ(b t ) √ √ −4pqQ(a t )Q(b t ),
I
µ 2 l − µ(1+) l l
µ+1 e e+µ l 2 ⎛ ⎞ (1 + ) × Iµ−1 ⎝2µ l ⎠ , l
Ps (e|t ) =
√ 2 where Q(x) = (1/ 2) x e−t /2 dt is the Gaussian Q1 1 6 function, p = 1 − MI , q = 1 − MQ , a = (M2 −1)+(M 2 −1)2 ,
B. -µ Fading Distribution µ+1 2
where P(e|t ) is the conditional symbol error rate for additive white Gaussian channels and p (t ) is the the PDF of the output SNR. The exact conditional SER for M = MI × MQ rectangular QAM is given as [11]
(8)
t
Thus, using (8) in (7) ASER for rectangular QAM can be evaluated. 3) Average Bit Error Rate for Some Binary Modulation Schemes: A unified conditional bit error rate (BER) expression for coherent and non-coherent binary modulation schemes can be given as [13] Pb (e|t ) =
√ AQ(B t ),
(9)
where A and B are modulation dependent parameters whose values for a number of modulation schemes are listed in [13]. Now using (3), (6) and (9) in (4) an expression for the average bit error rate (ABER) can be derived as 2B2 A 1 B2 1 + 1 Pb (e) , (10) 4 3 2 3 where 1 (·) is defined in (8) 4) Average Symbol Error Rate of coherent DE-QPSK and /4-QPSK Modulation Schemes: Both coherent, DE-QPSK and /4-QPSK modulation schemes have same conditional SER expression, which is given as [10, (8.39)] √ √ Q( t ) Q4 ( t ) √ √ − Q2 ( t ) + Q3 ( t ) − Ps (e|t ) = 8 , 2 2 (11) Following the steps discussed for rectangular QAM in Section III-A2, ASER expressions for these modulation schemes can be obtained.
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IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 1, NO. 4, AUGUST 2012
Ps (e)
1 a2 1 2a2 1 b2 1 2b2 1 a2 + b2 + 1 + 1 1 + q 1 − pq 1 6 2 2 3 6 2 2 3 36 2
2 2 2 2 2 2 3a + 4b 4a + 3b 2a + 2b 1 1 1 + 1 + 1 , + 1 12 6 12 6 4 3 p
0
0
10
10
−1
10
−2
−2
4x2−QAM
10
L=2 −4
10
−5
10
−6
0
=1, Approx. =1, Exact =21/5, Approx. L=3 =21/5, Exact 5
−3
10
L=2
−4
10
L=3 L=4
−5
10
−6
10 15 20 25 Average input SNR per branch (dB)
30
ASER for 4 × 2-QAM for -µ fading with = 2, µ = 1.5.
Fig. 1.
L=1
10
b
ABER, P (e)
s
ASER, P (e)
BPSK
L=1
10
−3
Analytical (Approx.) Simulation (Exact)
−1
10
10
(7)
10
Fig. 3.
0
5
10 15 20 25 Average input SNR per branch (dB)
30
ABER for BPSK for -µ fading with = 2, µ = 0.5.
0
10
2) Average Symbol Error Rate for Rectangular QAM: Using (12), (5) and (6) in (4), an expression for the ASER can be obtainedas shown in (13) at the top of the next page, where 2 () = 0 e−t p−µ (t )dt . Applying [12, (29.3.81)] and simplifying, a solution for the integral 2 () can be given as Lµ(1 + ) Lµ Lµt − 2 () = e Lµ(1+)+t . (14) Lµ(1 + ) + t
−1
10
L=1
−2
ASER, Ps(e)
10
−3
10
L=2
−4
10
4x2−QAM
−5
10
−6
10
Fig. 2.
0
Analytical (Approx.) Simulation (Exact) 5
L=3 L=4
10 15 20 25 Average input SNR per branch (dB)
30
ASER for 4 × 2-QAM for -µ fading with = 2, µ = 1 and = 1.
B. -µ Fading Channels 1) PDF of MRC Output SNR: Sum of L independent identically distributed (-µ)-square variates can be shown to be (-Lµ)-square distributed [1]. Hence, the PDF of t can be obtained by first obtaining the PDF of (-µ)-square variate from (2), scaling it by the factor Es /N0 and then substituting Lµ in place of µ, in the resulting expression. Thus, the final expression for the PDF of t can be given as Lµ+1
Lµ−1 t + 1 + 2 t 2 −Lµ (1+) t p−µ (t ) = Lµ e t t × ILµ−1 2Lµ (1 + ) . (12) t
Following the steps discussed for obtaining ASER over -µ fading channels, ASER for binary, coherent DE-QPSK and /4-QPSK modulation schemes can be obtained for − µ fading channels. IV. N UMERICAL R ESULTS AND D ISCUSSION Derived approximate expressions for ASER have been evaluated numerically and plotted for different modulation schemes. In Fig. 1 curves for ASER vs. average SNR per branch have been plotted for rectangular 4 × 2 QAM signal for -µ channels with varying and L. For the purpose of comparing the obtained results with the available results in [5, (18), (22), (23)] we have taken = 2 and µ = 1.5 in this figure. The ASER decreases with increase in L for a given . For example, for = 1, ASER for L = 3 is less than the ASER for L = 2. An improvement of ≈ 4 dB at an ASER of 10−4 can be observed from this figure. However, for a given L ASER degrades fast with increase in . Curves in this figure have been compared with the available exact results in [5] which are closely matching. It can be mentioned here that although exact ASER expression is available in [5] its evaluation is not straightforward since it is in the form of special functions (Appell’s and Lauricella’s
DIXIT and SAHU: PERFORMANCE OF L-BRANCH MRC RECEIVER IN -µ AND -µ FADING CHANNELS FOR QAM SIGNALS
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1 a2 1 2a2 1 b2 1 2b2 1 a2 + b2 + 2 + 2 2 + q 2 − pq 2 6 2 2 3 6 2 2 3 36 2
2 2 2 2 2 2 3a + 4b 4a + 3b 2a + 2b 1 1 1 + 2 + 2 , + 2 12 6 12 6 4 3
Ps (e)
p
ABER of 10−4 it can be observed that -µ channel requires an SNR ≈ 3 dB less than the SNR required in -µ channel. The SNR advantage of the -µ channels may be attributed to the presence of LOS component in the channel model.
0
10
Analytical (Approx.) Simulation (Exact)
−1
10
(13)
BPSK L=1
V. C ONCLUSION
b
ABER, P (e)
−2
10
−3
10
L=2
−4
10
L=3
−5
10
L=4
−6
10
Fig. 4.
0
5
10 15 20 25 Average input SNR per branch (dB)
30
ABER for BPSK for -µ fading with = 2, µ = 1.
TABLE I A BSOLUTE R ELATIVE D IFFERENCE BETWEEN E XACT AND A PPROXIMATE VALUES OF ASER FOR 4 × 2-QAM SYSTEM WITH = 1, = 2,µ = 1.5. SNR (dB) 0
5
10
20
L 1 2 3 1 2 3 1 2 3 1 2 3
Exact 0.599308 0.466075 0.370893 0.38131 0.203562 0.112786 0.141513 0.0309762 0.00721347 0.00163664 4.90839 × 10−6 1.63384 × 10−8
ASER Approximate 0.568851 0.481559 0.405226 0.406097 0.23881 0.138049 0.165575 0.0381659 0.00874488 0.00198067 5.66431 × 10−6 1.80044 × 10−8
Relative Diff.(%) 5.08 3.32 9.27 6.50 17.32 22.40 17.00 23.21 21.23 21.02 15.40 10.20
hypergeometric functions) which cannot be evaluated using software packages such as MATLAB and MATHEMATICA as these special functions are not available as built-in functions. The expression obtained here although approximate is easy to handle and the result is very close to the exact result. Table I tabulates the absolute relative difference percentage defined as |Exact−Approximate| × 100 as a function of SNR and L. Both Exact Fig. 1 and the table demonstrate the closeness between these results. In Fig. 2 ASER curves for -µ fading channels for 4 × 2-QAM signal are shown. A trend similar to the curves in Fig.1 can be observed for these set of curves also. In√ Figs. 3 and 4 ABER curves for BPSK modulation (A = 1, B = 2) for -µ and -µ fading channels, respectively, have been shown. On comparing the curves in Figs. 3 and 4, for L = 3, for an
ASER performance analysis of a MRC receiver in µ and -µ fading channels is presented. Approximate but highly accurate ASER expressions for L-branch MRC receiver with general order rectangular QAM modulation schemes are presented for -µ and -µ fading channels. These expressions are valid for all values of fading parameters. Numerical results validate the accuracy of the ASER expressions. Obtained results compared with the available results in literature and Monte Carlo simulation results, have been found to be closely matching. R EFERENCES [1] M. D. Yacoub, “The -µ distribution and the -µ distribution,” IEEE Antennas. Propag. Mag., vol. 49, no. 1, pp. 68–81, Feb. 2007. [2] D. B. da Costa and M. D. Yacoub, “Moment generating functions of generalized fading distributions and applications,” IEEE Commun. Lett., vol. 12, no. 2, pp. 112–114, Feb. 2008. [3] N. Y. Ermolova, “Moment generating functions of the generalized -µ and -µ distributions and their applications to performance evaluations of communication systems,” IEEE Commun. Lett., vol. 12, no. 7, pp. 502–504, July 2008. [4] N. Y. Ermolova, “Useful integrals for performance evaluation of communication systems in generalised -µ and -µ fading channels,” IET Commun., vol. 3, no. 2, pp. 303–308, Feb. 2009. [5] K. Peppas, F. Lazarakis, A. Alexandridis, and K. Dangakis, “Error performance of digital modulation schemes with MRC diversity reception over -µ fading channels,” IEEE Trans. Wireless Commun., vol. 8, no. 10, pp. 4974–4980, Oct. 2009. [6] M. Milisic, M. Hamza, and M. Hadzialic, “Outage and symbol error probability performance of L-branch maximal-ratio combiner for generalized -µ fading,” in Proc. 2008 International Symposium ELMAR, pp. 10–12. [7] M. Milisic, M. Hamza, N. Behlilovic, and M. Hadzialic, “Symbol error probability performance of L-branch maximal-ratio combiner for generalized -µ fading,” in Proc. 2009 IEEE VTC. – Spring, pp. 1–5. [8] M. Chiani, D. Dardari, and M. K. Simon, “New exponential bounds and approximations for the computation of error probability in fading channels,” IEEE Trans. Commun., vol. 2, pp. 840–845, July 2003. [9] J. G. Proakis, Digital Communications, 4th edition. McGraw-Hill, 2001. [10] M. K. Simon and M.-S. Alouini, Digital Communication over Fading Channels, 2nd edition. Wiley, 2005. [11] N. C. Beaulieu, “A useful integral for wireless communication theory and its application to rectangular signaling constellation error rates,” IEEE Trans. Commun., vol. 54, no. 5, pp. 802–805, May. 2006. [12] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th edition. Dover, 1970. [13] N. C. Sagias, D. A. Zogas, and G. K. Kariaginnidis, “Selection diversity receivers over nonidentical Weibull fading channels,” IEEE Trans. Veh. Technol., vol. 54, no. 6, pp. 2146–2151, Nov. 2005.
