A novel approach for performance analysis of wireless receiver with selection combining in Weibull Fading channel.

Malhotra, Jyoteesh ; Sharma, Ajay K. ; Kaler, R.S. 等

Introduction

Experimental data supporting the use of the Weibull distribution as a statistical model that better describes the actual short-term fading phenomenon over outdoor as well as indoor wireless channels has been reported by many researchers [1]-[4]. In addition, recent measurements in the cellular band carried out in Rio de Janeiro, Brazil, show that the variability of the signal on small areas generally follows a Weibull distribution [5]. Since then, the Weibull distribution has attracted much attention among the radio community. In particular, a closed-form expression for the moment generating function (MGF) of the Weibull random variable (RV) was obtained in [6] when the Weibull fading parameter is restricted to integer values. Another expression for the MGF of diversity receiver in multichannel Weibull fading was derived in [7]. Both expressions were given in terms of the Meijer's G function and were used for evaluating the performance of digital modulation schemes over Weibull fading channel. The closed-form expressions provided in [6, 7] despite being the first of their kind in the open literature, suffer from a major drawback. The expressions involving Meijer's G special function can be evaluated by itself using the modern mathematical packages such as Mathematica and Maple but these packages fail to handle integrals involving such functions and lead to numerical instabilities and erroneous results for higher values of fading parameters. This renders the expressions impractical from the ease of computation point of view as performance evaluation involves integration of this special function. Hence, it is highly desirable to find alternative closed-form expressions for the MGF of the Weibull RV that are simple to evaluate and at the same time can be used for arbitrary values of fading parameter. We choose to use the moment based PA technique [8] to find simple closed-form expression for the MGF for both single and multichannel receivers. Using PA to approximate MGFs through the knowledge of moments was introduced in [9] and [10], and the technique was applied for solving problems in communications over fading channels for the first time in [11]. Computation of the outage probability and average bit error rate using this technique was also done in [12] and [13], respectively. In this paper, we will use the moments based approach for obtaining closed form rational expressions of MGF [14] to evaluate the performance in terms of the outage probability and the error rate for different multilevel modulation schemes. This versatile and unified approach has been used to evaluate the performance of both single and multichannel reception. Comparisons with computer simulations are also established for the different numerical evaluations in this work.

The rest of the paper is organized as follows. In the following two sections, we present our system model and illustrate how the moments based approach can be efficiently used to obtain the MGF of the output SNR in single and multichannel reception. Section 4 details the performance analysis of the system in terms of outage probability and average error rate for both single channel and multichannel receiver with selection combining. The numerical and simulation results are presented and discussed in Section 5 before the paper is finally concluded in Section 6.

System and Channel Model

We consider signal transmission over slow, flat-frequency Weibull fading channel. The baseband representation of the received signal is given by y = sx + n, where s is the transmitted baseband signal, x is the channel envelope which is Weibull distributed, and n is the additive white Gaussian noise (AWGN). The PDF of the Weibull RV x at the receiver is given by [6]

[f.sub.x](x) = [cx.sup.c-1]/[gamma] exp (-[x.sup.c]/[gamma]) (1)

where the index c is called the Weibull fading parameter and [gamma] is a positive scale parameter given by [([OMEGA]/[GAMMA](1 + 2/c)).sup.c/2], where [GAMMA](.) is the Gamma function and [OMEGA] is the mean square value of RV. The Weibull fading parameter can take values between 0 and [infinity]. The nth moment of x can be obtained from (1) as

E([x.sup.n]) = [[gamma].sup.n/c][GAMMA] (1 + n/c] (2)

In general, the performance of any wireless receiver, in terms of bit error rate and signal outage, will depend on the statistics of the output signal-to-noise ratio (SNR), which is given as [chi square][S.sub.N] where [S.sub.N] is average signal and noise power ratio. Assuming that both the average signal and noise powers are unity, then the SNR will be equal to the squared fading channel amplitude i.e. SNR RV is given by the transformation y = [chi square]. One of the interesting properties of the Weibull RV with parameters (c, [gamma]) is that raising it to the kth power results in another Weibull RV with parameters (c/k, [gamma]). Hence, for a fading channel having a Weibull distributed amplitude with parameters (c , [gamma]), the SNR y is also Weibull distributed with parameters (c/2, [gamma]). Now, the cumulative distribution function (CDF) of Weibull RV is given as [14]

[F.sub.Y](y) = 1 - exp (-[y.sup.c/2]/[gamma]) (3)

Using(3), it can be shown that the CDF for the K-branch independent multichannel selection combining (SC) receiver is [17, chap.7]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

From (4) and [16, eq. (3.381)] the nth moment of SC combiner output SNR is derived as

E[[Y.sup.n.sub.SC]] = K[[gamma].sup.2n/c][GAMMA](2n/c + 1) [K-1.summation over (k = 0)](-1).sup.k] (K - 1 k) [(k + 1).sup.-2n/c-k] (5)

MGF Using Pade Method

The MGF of an RV x > 0 is given as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where E(.) is the Expectation operator. Based on the discussion presented in first Section, it is required to find an alternative closed-form expression for the MGF which is simpler to use in computations and at the same time valid for any arbitrary value of fading parameter. We will find the alternative closed form rational expression using (2) & (6). Using the Taylor series expansion of [e.sup.sx], the MGF can be expressed in terms of a power series as

[M.sub.x](s) = [[infinity].summation over (n = 0)] [(-1).sup.n]/n! E([x.sup.n]).[s.sup.n]

= [[infinity].summation over (n = 0)] [c.sub.n][s.sup.n] (7)

The infinite series in (7) can be efficiently approximated by a rational function using the PA method. In particular, the one-point PA of order (M-1 / M) is defined from the series in (7) as a rational function given by

[M.sub.x](s)[equivalent] [M-1.summation over (i = 0)] [a.sub.i][s.sup.i]/[M.summation over (j = 0)] [b.sub.j][s.sup.j] (8)

where [a.sub.i] and [b.sub.j] are the coefficients such that

[M-1.summation over (i = 0)] [a.sub.i][s.sup.i]/[M.summation over (j = 0)] [b.sub.j][s.sup.j] = [2 M-1.summation over (n = 0] [c.sub.n][s.sup.n] + O([s.sup.2M]) (9)

where O([s.sup.2M]) representing the higher order terms. The coefficients [b.sub.j] can be found using (assuming [b.sub.0] = 1) following equations

[M.summation over (j = 0)] [b.sub.j][c.sub.M-1-j+k] = 0 0 [less than or equal to] k [less than or equal to M (10)

The above equations form a system of M linear equations for the M unknown denominator coefficients in(8). This system of equations given in (10) can be uniquely solved, as long as the determinant of its Hankel matrix is nonzero [9]. The choice of the value of M is indeed a critical issue, as it represents a tradeoff between the accuracy and complexity of the system of equations to be solved. It is described in [9] that there exist a value of M above which Hankel matrix become rank deficient. After solving for the values of [b.sub.j], the set [a.sub.i] can now be obtained from following set of equations

[a.sub.i] = [c.sub.i] + [min(M ,i).summation over (n = 1)] [b.sub.i][c.sub.i-n] = 0 0 [less than or equal to] i [less than or equal to] M - 1 (11)

Having obtained the coefficients of denominator and numerator polynomials, an expression for the MGF of the output SNR can be derived in rational function form. We are now ready to present two of the most important performance measures namely, the outage probability and the average BER for different signaling schemes.

Performance Analysis

In this section the performance analysis of various classes of receivers operating over Weibull fading channel is presented. Both Single Channel and Multichannel receivers using SC have been used for analysis.

MGF in Single Channel Receiver

The closed-form rational expressions of MGF have been evaluated for arbitrary values of Weibull fading parameter for the single channel receiver. Interestingly, in special case of c = 2, Hankel Matrix is rank deficient except for D = 1, the only unknown coefficient [b.sub.1] can be easily found to be 1.The MGF found in this case is thus given by

[M.sub.x](s) = 1/(1 + [gamma]s) (12)

The MGF of output SNR obtained above is in closed form rational expression form, exactly matches the expression of MGF of SNR given in [14] for Rayleigh faded envelope. Thus, the PA method leads to an exact expression for the special case of c = 2, which verifies its accuracy. Now, we will obtain the rational expressions of the output SNR MGF for non-integer (c = 2.5) and higher integer (c = 4) values of fading parameter in order to demonstrate the versatility in obtaining computationally simple expressions. The rational form expressions of MGF for c = 2.5 and c = 4 have been derived in (13) & (14).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

MGF in Selection Combining Receiver

The work has been extended to find out MGF of the SC output SNR using (5) &(6). For fading parameter c = 2, the simple closed form expressions of MGF with dual and triple SC have been derived as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

Further, we have obtained the rational expressions of MGF for selection diversity combining receiver. Equation (17), (18) are the rational expressions for the MGF for c = 2.5 and c = 4 with dual SC, while (19) & (20) for the triple SC. Since the rational expression obtained here are very simple and does not have any restriction on the values of fading parameter, it is now easy to obtain performance results for integer as well as non-integer values of fading parameter.

MGF of Dual SC for c = 2.5

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

MGF of Dual SC for c = 4

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

MGF of Triple SC for c = 2.5

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

MGF of Triple SC for c = 4

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

Average Bit Error Rate

For Gray encoded Multilevel Differential Phase Shift Keying (MDPSK) the conditional BER as given in [15]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

where [varies] is the instantaneous SNR , L is the number of symbols. Using MGF approach, the average BER will be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

For Gray encoded Multilevel Quadrature Amplitude Modulation (MQAM), the conditional BER as given in [14] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

where [g.sub.qam] = 3/2(L-1)

The average BER of MQAM can be obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

Similarly for employing Multilevel Phase Shift Keying (MPSK) with coherent detection, the average BER can be obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

where [g.sub.psk] = [sin.sup.2]([pi]/L)

By using the above cited method, average BER performance for the diversity combining receiver can be obtained using the MGF expressions derived in (15) to (20) of the SC output SNR.

Outage Probability

The outage probability (OP) is defined as the probability that the SNR drops below a certain threshold, [xi] viz

[F.sub.out]([xi]) [equivalent] P(SNR < [xi]) (26)

Using MGF approach [Chapter 1, 14] the outage probability of single channel can be computed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

where [M.sub.x] (s,[gamma]) is the MGF of the output SNR and a is a properly chosen constant in the region of convergence in the complex s-plane. Since [M.sub.x] (s, [gamma]) is given in terms of a rational function, we can use the partial fraction expansion of [M.sub.x](s,[gamma])/s to evaluate outage probability i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

where [p.sub.i] are the [N.sub.p] poles of rational function in s with [[lambda].sub.i] its residues. Each term inside the summation in (28) represents a simple rational function form.

The outage probability in case of K-branch selection combining is given by [14, Sec.1.1.2]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([xi]) is the outage probability which can be obtained from its MGF by inverse Laplace transform as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

Thus, we have obtained computationally simple rational expressions for the MGF of the output SNR with and without diversity combining. In some of the rational expressions, closed form can also be found as it is equivalent to the problem of finding the inverse Laplace transform. Using these novel expressions performance stable evaluation results can be easily obtained. Based on the analysis presented in this section numerical and simulation evaluation results are presented next.

Numerical and Simulation Results

Outage probability and Average BER of wireless receiver through Weibull fading channel have been numerically evaluated and compared with simulation results.

Outage Probability

Figure 1 depicts the outage probability [F.sub.out] in single channel reception versus the threshold [xi], normalized by scaling parameter [gamma]. The outage probability results obtained from (27) and Monte Carlo simulation are shown. It is evident from figure that there is perfect agreement between both results. It is also observed that as the fading parameter c is increased from 2 to 4 in the figure, the degree of fading severity decreases. The numerical results of OP in multichannel reception based on Dual and Triple SC obtained using (30) are presented along with simulated results in Figure 2. It is apparent that the outage performance of the system becomes significantly better with the deployment of higher order diversity combining.

Average Bit Error Rate

We have chosen three multilevel modulations to illustrate the average BER performance of single and multichannel wireless receivers versus the average SNR per bit. Firstly, Figure 3 & 4 shows ABER for 16-DPSK in single and triple channel selection combining, respectively. Figure 5 & 6 shows ABER for 16-QAM in single and triple channel selection combining, respectively. Figure 7 & 8 depicts the ABER results for 16-PSK in single and triple channel selection combining, respectively. As in the case of the outage probability, the ABER improves as the fading parameter is increased from 2 to 4. The performance evaluation of the single and selection diversity receiver is also done for the non-integer value i.e. c=2.5. Selection combining mitigates the fading effect, which is indicated by the reduction in error probability at any fixed value of average SNR in all three modulation formats. MQAM has least probability of error than both MDPSK and MPSK in the Weibull fading scenario. Coherent detection gives better ABER performance than differential detection for the same SNR as expected. As evident from the figures there is perfect match between the numerical results and simulation results for the different modulation formats in Weibull fading channel with arbitrary values of fading parameter.

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[FIGURE 2 OMITTED]

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This can be deduced from the results that the moment based approach can be used to give very accurate estimate of the performance in terms of outage and ABER for both integer and non-integer values of Weibull fading parameter. If the accuracy is not satisfactory in some cases for higher values of SNR, it is always possible to choose a larger value of M to enhance accuracy as long as the Hankel matrix is not rank deficient. Thus, alternative simple to evaluate rational expressions of MGF resulted in unified performance analysis of multichannel reception employing selection combining.

Conclusions

In this paper, we evaluated the performance of the wireless receiver operating over Weibull fading channel using moment based approach. A rational form representation of the output SNR MGF was first obtained for both single channel and multichannel receivers with selection combining. It was then used to quantify the performance in terms of the outage probability and the average BER for different multilevel modulations. Numerical as well as simulation results were presented to complement the analytical content of the paper. Several examples have been presented to corroborate the accuracy of newly obtained expressions with the previously published closed-form expression, which shows perfect agreement. We also presented a new set of results for the cases of non-integer values of the Weibull fading parameter. Thus, moment based approach proves to be an invaluable tool for obtaining computationally simple and accurate expressions for the MGF, which can be used for further analysis.

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Jyoteesh Malhotra (1), Ajay K. Sharma (2) and R.S Kaler (3)

(1) Lecturer, Dept. of ECE, Guru Nanak Dev University Regional Campus, Jalandhar, India E-mail: jyoteesh@rediffmail.com

(2) Professor, Dept. of CSE, National Institute of Technology (DU) Jalandhar, India, sharmaajayk@rediffmail.com (3) Professor, Dept. of ECE, Thapar University Patiala, India E-mail: rskaler@yahoo.com