Performance Analysis of L-Branch Scan-and-Wait Combining (SWC) over Arbitrarily Correlated Nakagami-m Fading Channels

# Performance Analysis of L-Branch Scan-and-Wait Combining (SWC) over Arbitrarily Correlated Nakagami-m Fading Channels

George C. Alexandropoulos, , P. Takis Mathiopoulos, , and Pingzhi Fan,  G. C. Alexandropoulos is with Mathematical and Algorithmic Sciences Lab, France Research Center, Huawei Technologies Co. Ltd., 92100 Boulogne-Billancourt, France (e-mail: george.alexandropoulos@huawei.com).P. T. Mathiopoulos is with Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, 15784 Ilissia, Athens, Greece (e-mail: mathio@di.uoa.gr).P. Fan is with Institute of Mobile Communications, Southwest Jiaotong University, Chengdu, Sichuan 610031, PR of China (e-mail: p.fan@ieee.org).
###### Abstract

The performance of -branch scan-and-wait combining (SWC) reception systems over arbitrarily correlated and not necessarily identically distributed Nakagami- fading channels is analyzed and evaluated. Firstly, a fast convergent infinite series representation for the SWC output signal-noise ratio (SNR) is presented. This expression is used to obtain analytical expressions in the form of infinite series for the average error probability performance of various modulation schemes for integer values of as well as the average number of paths estimation and average waiting time (AWT) of -branch SWC receivers for arbitrary values of . The numerically obtained results have shown that the performance expressions converge very fast to their exact analytical values. It was found that the convergence speed depends on the correlation and operating SNR values as well as the Nakagami -parameter. In addition to the analytical results, complementary computer simulated performance evaluation results have been obtained by means of Monte Carlo error counting techniques. The match between these two sets of results has verified the accuracy of the proposed mathematical analysis. Furthermore, it is revealed that, at the expense of a negligible AWT, the average error probability performance of SWC receivers is always superior to that of switched-and-examine combining receivers and in certain cases to that of maximal-ratio combining receivers.

Correlated fading, error probability, Nakagami- distribution, scan and wait, switched diversity, waiting time.

## I Introduction

Spatial diversity techniques play an important role in current mobile broadband systems as efficient means of mitigating channel fading due to multipath and shadowing [1, 2]. The vast majority of these techniques requires dedicated channel estimation and matched filtering for every diversity branch which, unfortunately, increase their implementation complexity [3, 4]. To reduce complexity and processing power consumption, there has been in recent years considerable interest in spatial diversity techniques that utilize only a subset of the available diversity branches, eg generalized selection combining (GSC) [5], minimum selection GSC [6], switched diversity [7, 8, 9, 10, 11], minimum estimation and combining GSC [12] as well as adaptive GSC [13]. It is noted that the high reliability of wireless connections is a crucial requirement especially in emerging technologies optimized for low-cost and low-power consumption [14, 15], such as low-rate wireless personal area networks and sensor networks [16]. For such kind of technologies, spatial diversity with a reduced number of radio frequency (RF) chains is an appropriate and promising solution.

Two of the most popular diversity techniques that utilize a single RF chain are switch-and-stay combining (SSC) [3, 7, 11] and switch-and-examine combining (SEC) [8, 17]. In multi-branch SSC, the transmitter/receiver switches to, and stays, with the next available diversity branch regardless of its signal-to-noise ratio (SNR), when the instantaneous SNR on the current branch becomes unacceptable, ie lower than a preset switching threshold. For the SEC, the combiner first examines the next branch’s SNR and switches again only if this SNR is unacceptable. In case where the instantaneous SNR of all available branches is lower than the preset switching threshold, the combiner either uses the last examined branch or switches back to the first branch for the next operation slot. An alternative form of SEC, termed as scan-and-wait combining (SWC), suitable for the sporadic communication of delay-tolerant information over wireless networks, such as wireless ad-hoc and sensor networks, was presented in [10]. With SWC, if all the available diversity paths fail to meet a predetermined minimum quality requirement, the system waits for a certain time period and restarts the switch-and-examine process. This scanning followed by waiting can then be repeated until a path with acceptable quality is found. It was shown in [10] that, for a fixed average number of channel estimates per channel access, SWC outperforms SSC and SEC at the expense of a negligible time delay.

• The derivation of a generic analytical expression in the form of fast convergent infinite series for the joint probability density function (PDF) of AC-NNID Gamma fading random variables (RVs). This expression is then conveniently used to obtain an analytical expression for the -branch SWC output SNR.

• Fast convergent infinite series representations for the AWT and the average number of path estimations (ANPE) of -branch SWC receivers with for arbitrary values of the Nakagami -parameter as well as the average error probability of various modulation schemes of the same receivers for integer values of are derived. Furthermore, a novel analytical expression for the ANPE of -branch SEC receivers for arbitrary-valued -parameter is presented.

To verify the validity of the analytical approach, various numerically evaluated and computer simulation performance evaluation results will be presented and compared. In addition, extensive error probability performance comparisons with -branch SEC and maximal-ratio combining (MRC) receivers are included and discussed.

The reminder of this paper is organized as follows. After this introduction, Section II presents the derivation of the analytical expression for the -branch SWC output SNR. In Section III, the various analytical expressions obtained for the performance of SWC receivers are presented. The performance evaluation results are presented and discussed in Section IV. In Section V the conclusions of the paper can be found.

## Ii Statistics of the SWC Output SNR

A SWC receiver with antenna branches receiving digitally modulated signals transmitted over a slow varying and frequency nonselective Nakagami- fading channel is considered. Let and denote the vectors with the instantaneous received SNRs and the predetermined SNR thresholds at the branches, respectively. For the considered fading model, is a Gamma distributed RV with marginal cumulative distribution function (CDF) given by

 Fgℓ(g)=γ(m,mg¯¯¯gℓ)/Γ(m) (1)

where the parameter relates to the fading severity, denotes the average received SNR at the th branch, is the lower incomplete Gamma function [31, eq. (8.350/1)] and is the Gamma function [31, eq. (8.310/1)]. We consider the general case where ’s are arbitrarily correlated with correlation matrix (CM) . This matrix is symmetric, positive definite, and given by , , and , where is the correlation coefficient between and [20, eq. (9.195)]. With SWC, in the guard period between every two consecutive time slots for data transmission [10], the receiver measures of branch and compares it to . If , then branch is selected for information reception in the upcoming time slot. However, if , the receiver switches to branch , measures and compares to . This procedure is repeated until either an acceptable branch for information reception is found or all available branches have been examined without finding any acceptable one, ie . In the latter case the receiver informs the transmitter through a feedback channel not to transmit during the upcoming time slot and to buffer the input data for a certain waiting period of time. After that waiting period, the system re-initiates the same procedure. Based on this mode of operation, the PDF of the SWC output SNR is given by [10, eq. (1)]

 (2)

where denotes the joint CDF of and is the conditional PDF of the truncated (above ) given that , , , . For the conditional PDF simplifies to

 fTg1(g)={fg1(g),g≥gT10,otherwise (3a) whereas for ℓ=2,3,…,L it can be expressed as (3b)

In (3), is the marginal PDF of obtained by differentiating (1) [7, Tab. I], denotes multiple integrations and is the joint PDF of .

Clearly from (2) and (3), in order to analyze the performance of -branch SWC receivers over AC-NNID Nakagami- fading channels, analytical expressions for the joint statistics of are needed. A generic analytical expression for the joint CDF of , that encompasses various cases for , and the form of [21, 22, 23, 24, 25, 26, 27, 28], was presented in [30, eq. (6)]

 Fg(L)(g(L))=∞∑i(N)A(m,Σ,i(N))L∏ℓ=1γ(κℓ(m),ξℓ(m,Σ)gℓ¯¯¯gℓ) (4)

where , with , represents the multiple infinite series111In practice, to numerically evaluate (4), minimum numbers of terms are selected in the summations leading to a certain accuracy. An upper bound for the resulting truncation error is given by [30, eq. (7)]. In addition, [30] includes results with the minimum numbers of required terms in (4) for convergence to the sixth significant digit for various values of the involved parameters, eg , , , ’s and . and . Moreover, as shown in [30], and the real-valued scalar functions , and depend on the form of the CM and the fading parameter . For example, considering an exponential CM, ie with denoting absolute value, the joint CDF of is obtained: i) by substituting in (4) and

 A(m,Σ,i(N))=det(W)mΓ(m)∏Lℓ=1Wκℓ(m)ℓ,ℓL−1∏j=1W2ijj,j+1ij!Γ(ij+m) (5)

with denoting the th element of . Also, ii) by setting in (4) and (5) , for and ; as well as iii) by substituting in (4) . Differentiating (4) times and using [31, eq. (3.381/1)], yields the following very generic analytical expression for the joint PDF of :

 fg(L)(g(L))=∞∑i(N)A(m,Σ,i(N))L∏ℓ=1[ξℓ(m,Σ)¯¯¯gℓ]κℓ(m)gκℓ(m)−1ℓexp(−ξℓ(m,Σ)gℓ¯¯¯gℓ). (6)

Substituting in (3a) for as well as (6) in (3b) for and using [31, eq. (3.381/1)] to solve the resulting multiple integrals, an analytical expression for the numerator of (2) is obtained. Using this expression and substituting (4) in the denominator of (2), a general analytical expression for the PDF of the SWC output SNR over AC-NNID Nakagami- fading channels is given by

 fgSWC(g)=⎡⎣1−∞∑i(NL)A(m,Σ(L),i(NL))L∏ℓ=1γ(κℓ(m),ξℓ(m,Σ(L))gTℓ¯¯¯gℓ)⎤⎦−1×⎧⎨⎩(m¯¯¯g1)mgm−1Γ(m)exp(−mg¯¯¯g1)+L∑ℓ=2∞∑i(Nℓ)A(m,Σ(ℓ),i(Nℓ))[ℓ−1∏j=1γ(κj(m),ξj(m,Σ(ℓ))gTj¯¯¯gj)]×[ξℓ(m,Σ(ℓ))¯¯¯gℓ]κℓ(m)gκℓ(m)−1exp(−ξℓ(m,Σ(ℓ))g¯¯¯gℓ)⎫⎬⎭. (7)

This expression is valid for and , and depends on the form of the CM of the instantaneous SNRs of the diversity branches [30]. For example, for an arbitrary CM , while for and exponentially correlated , . To evaluate (7), infinite series in the numerator and in the denominator need to be evaluated, for which minimum numbers of terms are selected in practice. It is also noted that (7) can be easily modified to account for the cases where and/or for any . In such cases, the first and/or the th multiple infinite series term in the numerator of (7) should be set to .

## Iii Performance Analysis

The previously derived formulas will be used to obtain analytical expressions for the performance of -branch SWC receivers, in terms of AWT, ANPE and average error probability, over AC-NNID Nakagami- fading channels.

### Iii-a AWT and ANPE

To quantify the complexity and required processing power consumption as well as the delay of SWC receivers, the following performance metrics are considered [10]: i) The ANPE, , before channel access; and ii) The AWT, , in terms of the number of coherence times that the system has to wait before an acceptable path is found and transmission occurs. By substituting (1) and (4) into [10, eq. (28)], the following general analytical expression for can be obtained

 ¯¯¯¯¯Ne=¯¯¯¯¯NSEC[1−Fg(L)(g(L)T)]−1 (8)

where represents ANPE of SEC, which is given by

 ¯¯¯¯¯NSEC=L∑ℓ=1ℓPr[NSEC=ℓ]. (9)

In (9), denotes probability and is a discrete RV representing the number of path estimations for SEC. By using (1) and (4), and after some straightforward algebraic manipulations, can be expressed as

 ¯¯¯¯¯NSEC=1+L−1∑ℓ=1Fg(ℓ)(g(ℓ)T). (10)

Furthermore, by substituting (4) into [10, eq. (16)], a generic analytical expression for is derived as

 ¯¯¯¯¯Nc=Fg(L)(g(L)T)[1−Fg(L)(g(L)T)]−1. (11)

### Iii-B Average Error Probability

The average error probability performance of digital modulation schemes of -branch SWC receivers over fading channels can be evaluated using

 ¯¯¯¯Pe=∫∞0Pcp(g)fgSWC(g)dg (12)

where depends on the modulation scheme [1, Chap 5], [20, Chap 8]. For a great variety of such schemes, where are constants and denotes the Gaussian -function [20, eq. (4.1)]. For example, and leads to binary phase-shift keying (BPSK) whereas, and to -ary pulse amplitude modulation (-PAM). In addition, a tight approximation for ABEP of rectangular -ary pulse quadrature modulation (-QAM) can be obtained with and . By substituting (2) into (12), the average error probability of various modulation schemes of -branch SWC receivers over fading channels can be expressed as

 ¯¯¯¯Pe=A1−Fg(L)[g(L)T]L∑ℓ=1∫∞gTℓQ(√Bg)fTgℓ(g)dg. (13)

To evaluate the average error probability for AC-NNID Nakagami- fading channels, we first substitute and (6) into (3a) and (3b), respectively, and then into (13). To this end, integrals of the form , with and , need to be evaluated. As will be shown in the sequel, parameter depends on whereas, parameter is a function of , the average received SNRs and the correlation coefficients. Similar to [17, Sec III.A] and using integration by parts, can be rewritten as

 Iℓ(b,c)=c−b[Q(√BgTℓ)Γ(b,gTℓc)−Yℓ(b,c)√2π] (14)

where is the upper incomplete Gamma function [31, eq. (8.350/2)] and is given by

 Yℓ(b,c)=∫∞√BgTℓexp(−0.5g2)Γ(b,cg2B)dg. (15)

A closed-form solution for the integral in (15) can be obtained for integer-valued by using [31, eq. (8.352/7)] and [31, eq. (8.381/3)], as

 Yℓ(b,c)=0.5√B(b−1)!b−1∑n=0cnΓ(n+0.5,(0.5B+c)gTℓ)n!(0.5B+c)n+0.5. (16)

Finally, by substituting (16) into (14) and then in (13), yields a generic analytical expression for the average error probability performance of various modulation schemes over AC-NNID Nakagami- fading with integer , given by

 ¯¯¯¯Pe=A⎡⎣1−∞∑i(NL)A(m,Σ(L),i(NL))L∏ℓ=1γ(κℓ(m),ξℓ(m,Σ(L))gTℓ¯¯¯gℓ)⎤⎦−1{(m/¯¯¯g1)mΓ(m)I1(m,m¯¯¯g1)+L∑ℓ=2∞∑i(Nℓ)A(m,Σ(ℓ),i(Nℓ))[ℓ−1∏j=1γ(κj(m),ξj(m,Σ(ℓ))gTj¯¯¯gj)][ξℓ(m,Σ(ℓ))¯¯¯gℓ]κℓ(m)×Iℓ(κℓ(m),ξℓ(m,Σ(ℓ))¯¯¯gℓ)}. (17)

For the special case of dual-branch SWC receivers operating over AC-NNID Rayleigh channels, by setting and in (17) and then by substituting using [30, eqs. (11) and (12)]222A typo exists in the numerator in the right-hand side of the equality in [30, eq. (11b)], where needs to be replaced by . the resulting parameters as , and for and , the following analytical expression for the error probability is deduced

 ¯¯¯¯Pe=A[1−(1−√ρ)∞∑i1=0ρi1/2i1!Γ(i1+1)2∏ℓ=1γ(i1+1,gTℓ(1−√ρ)¯¯¯gℓ)]−1{1¯¯¯g1I1(1,1¯¯¯g1)+(1−√ρ)∞∑i1=0ρi1/2i1!Γ(i1+1)γ(i1+1,gT1(1−√ρ)¯¯¯g1)[1(1−√ρ)¯¯¯g2]i1+1×I2(i1+1,1(1−√ρ)¯¯¯g2)}. (18)

For this special case, the average error probability of various modulation schemes can also be evaluated using the MGF-based approach [20, Chap. 1] with the MGF expression given by [9, eq. (7)]. Although the latter expression was derived using the bivariate Rayleigh CDF given by [20, eq. (6.5)] while in obtaining (18) the alternative infinite series representation [21, eq. (4)] for this CDF was used, it will be shown in the following section, where the performance evaluation results will be presented, that both approaches yield identical average error probability performance.

## Iv Performance Evaluation Results

In this section, the analytical expressions (8), (11), (17) and (18) have been used to evaluate the performance of -branch SWC receivers over AC-NNID Nakagami- fading channels. Furthermore, and similar to [10], we have compared the average error performance of various modulation schemes of the considered receivers with that of SEC and MRC receivers. To this end, the analytical expressions [30, eq. (21)] and [18, eq. (11)] for the average symbol error probability (ASEP) performance of SEC and MRC, respectively, have been also evaluated. To verify the correctness of the proposed analysis, equivalent performance evaluation results obtained by means of Monte Carlo simulations will be also presented. For this, and in order to generate AC-NNID Gamma RVs with CM , the decomposition technique of [19] has been used. Accordingly, each of the AC-NNID Gamma RVs is obtained as a sum of AC-NNID Gaussian RVs with CM . Without loss of generality, for the performance results obtained we have considered: i) independent and identically distributed (IID) fading with as well as and ; and ii) AC-NNID fading characterized by exponential CMs with and the exponential power decaying profiles and , where is the power decaying factor.

In Figs 1 and 2, the average error performance of -branch SWC receivers is depicted as a function of the first branch average SNR, , and compared with the performance of SEC and MRC reception, respectively. Similar to [10], and for a fair comparison, in Fig 1 was set equal to and in Fig. 2 it was set equal , ie equal to the ANPE of MRC. In particular, in Fig 1 for , Nakagami- fading with and , and -PAM, we have used the ASEP expression [30, eq. (21)] for each value to compute the optimum SNR threshold minimizing SEC’s ASEP, as described in [30, Sec. III.B]. This was next substituted into (10) to evaluate that was then used in (8) to solve for of SWC. The computed values for the -branch SWC receivers the performance of which is shown in Fig 1 as a function of , are depicted in Fig 4. These values were utilized in (17) to evaluate ASEP of SWC. In Fig 2, for the SWC’s ABEP curves for and , Rayleigh fading (ie Nakagami with ), and BPSK modulation, (8) was first set to and for each value to solve for needed in (18) for -branch and in (17) for -branch SWC reception, respectively.

The resulting values are plotted in Fig 5 versus . In Fig 2, the ASEP with BPSK for MRC was evaluated using the MGF-based approach [20, Chap. 1] and [18, eq. (11)]. Similar to the approach for deriving the performance curves in Fig 2, in Figs 3 and 6 we have set (8) equal to and , ie was set equal to the ANPE of - and -branch MRC, respectively, to obtain ABEP of BPSK and -QAM modulation schemes of - and -branch SWC receivers for various values of and . We have selected only for the correlated case in Fig 3 and for every value of in Fig6.

For the numerical evaluation of (8), (11), (17) and (18) in the performance results shown in all figures, the minimum number of terms, , was used in all involved series of each multiple series term so that they converge with error less or equal to . The maximum value of was in all results for IID fading whereas, Table I summarizes the required number of to achieve this accuracy for the considered AC-NNID fading cases. As clearly shown in this table, values are quite low and increase as decreases and/or and/or increase.

The results in Figs 13 illustrate that error performance for all considered receivers improves with increasing and/or and/or decreasing . Furthermore, as shown in Fig 1, SWC outperforms SEC and this superiority is more pronounced when and/or and/or decrease. From the SWC’s ASEP curves in Fig 1, it is also evident from Fig 4 that increasing and/or and/or decreasing results in increasing . In addition, it can be observed that AWT increases as and/or decrease and/or increases. For example, considering IID fading channels, and for dB, and for and , respectively, while for the AC-NNID case and for and , respectively. Additional comparisons (not presented here due to space limitations) between SWC and SEC have further shown that the gains of SWC over SEC increase as decreases.

It can be also observed from Fig 2 that, as decreases and/or increases, the values for which the ABEP of SWC becomes better than the ABEP of MRC decrease. On the other hand, as shown in Fig 5, by increasing and/or , and/or decreasing , this increases for the SWC reception. Furthermore, it can be observed that the impact of increasing is more severe to MRC than SWC and this difference becomes larger as decreases. For example, for with - and -branch MRC reception, the gap between the considered IID and AC-NNID case is approximately dB whereas, for the case of SWC this gap decreases to approximately dB. For the ABEP results in Fig 2, it can be seen that the AWT of -branch SEC is independent of . Furthermore from the same figure, it can be seen that for the case of IID Rayleigh fading, and for and , respectively, while for the AC-NNID case for and for .

The impact of the modulation order and on the ABEP performance of -branch SWC receivers is demonstrated in Figs 3 and 6. As expected, increasing degrades ABEP and, for a given value, the impact of on ABEP increases with decreasing . As observed in Fig 6 for low values of and decreasing , ABEP improves when becomes large. Interestingly, as and increase as well as decreases, ABEP degrades with increasing .

## V Conclusion

In this paper, we have analyzed and evaluated the performance of -branch SWC receivers over AC-NNID Nakagami- fading channels. Fast convergent infinite series representations for the average error performance of various modulation schemes as well as the ANPE and AWT of the considered receivers were presented. As shown from the numerically evaluated results which have been verified by means of computer simulations, the superiority of error performance of SWC over SEC increases as fading conditions and correlated fading become more severe at the cost of negligible increasing AWT. It was also demonstrated that: i) the impact of increasing correlation is more severe to the average error performance of MRC than that of SWC; and ii) if the diversity reception system is resilient to some AWT, there are certain cases where the average error probability of SWC is superior to that of MRC.

## Acknowledgment

The work of P Fan was supported by Project (No --) and NSFC (No ).

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