A unified approach for the fading statistics of heterogeneous compound channels in transparent relay transmissions
© Kanjirathumkal et al.; licensee Springer. 2013
Received: 28 June 2013
Accepted: 19 November 2013
Published: 5 December 2013
In this paper, we address the challenging issue of multi-hop cooperative relaying over heterogeneous compound channels, where constituent channels can incorporate topographical variations and respective distribution changes (due to environmental factors). Existing methods in the literature address various issues pertinent to receiver design by assuming a homogeneous framework. Since real-time signal level monitoring at the relay node is impractical in transparent relaying, statistical knowledge of the end-to-end channel is critical for an energy-efficient system design. As the received signal quality and hence the signal-to-noise ratio (SNR) depend on the channel condition, an exact characterisation of the end-to-end compound channel is essential for the implementation of transparent relay networks in a particular coverage area. This demands a realistic heterogeneous compound channel model, having distinct component distributions, that can accommodate variations in per-hop channel distribution. Exact probability density function and cumulative distribution function of the end-to-end compound channel are derived in closed form, through an inverse Mellin transform approach. As there can be many combinations of heterogeneous channels, we propose a unified approach for deriving statistical properties of interest. Two typical cases of heterogeneous channels, Nakagami × Weibull and Rayleigh × Weibull, are considered to validate the proposed generic analytical approach. Based on the derived exact statistics of these models, performance metrics, such as coefficient of variation, average SNR, outage probability, and average SER, are evaluated. Through appropriate use of these metrics, optimal power boosting at the relays and quality stream selection for arbitrary diversity combiner units can be done. The pertinent analytical results are also validated through simulation studies to demonstrate the accuracy and applications of the results.
KeywordsMellin Transform Multi-hop non-regenerative relay Compound channels Weibull distribution Rayleigh distribution Nakagami-m distribution
With the possibility of providing enhanced network coverage as well as spatial diversity, wireless cooperative relaying is undoubtedly one of the most significant communication strategies today. Of the many possible schemes of relaying, transparent or amplify and forward (AF) relaying is usually preferred, as the processing burden on the relay is minimal, though the compounding channel effects due to cascaded channel gain multiplication is a serious issue. However, this demands sophisticated implementation techniques, as it can bring much more attenuation and uncertainty to the transmitted signal than other schemes do. Technologically, the extensive applications of wireless relay transmissions are impedimented by the undesirable features of the wireless transmission environments. One of the fundamental challenges to be addressed for wireless communication is the random fluctuation of the transmitted signal, which is mainly due to the multi-path effects and shadowing. In the context of multi-hop transparent relaying, additional impairments on the end-to-end channel from multiple relaying, with distinct per-hop distributions due to distinct environments, require special attention as it affects the system design and receiver performance. Appropriate power amplification by the intermediate relays is another major issue that has to be addressed in AF relay system. Amplification by the relay node based on respective signal strength measurement is impractical in AF systems, as this may increase overall delay and processing burden at the relay node. Hence, a statistical knowledge of the end-to-end channel under consideration is critical, to implement a robust communication system. The main objective of the proposed work was to provide a unified approach to extract the statistical properties of any compound channel, which will enable us to design links that account for per-hop channel distributions and to reap the benefits of diversity.
To address the issue of modeling random fluctuations of the transmitted signal, terrain dependent fading models for direct link transmissions, based on empirical study of the received signal, are available. Extending this to multi-hop scenario results in homogeneous compound channel models as in [1–3], and is more suitable for indoor applications. The double Rayleigh model analysis using Meijer G-function for multiple-input multiple-output (MIMO) systems in [4, 5] is often used in multi-hop relay transmissions where homogeneous product channel effects are experienced. As the relays are physically apart in multi-hop transmissions, spatial correlation can be neglected in modeling.
In outdoor multi-hop transmissions, the assumption of homogeneous scattering environment for all the constituent links is definitely an approximation; because in actual scenario, cascaded links are often characterised by heterogeneous environment due to scattering density variations. Hence, a more realistic channel model suitable for transparent relay transmissions is presented. The analysis is extended to it so as to make realistic prediction of signal variations at the receiver, and to facilitate appropriate detection strategies. Two cases of heterogeneous channel models, Nakagami × Weibull (NW) and Rayleigh × Weibull (RW), are considered to demonstrate the validity of the proposed generic approach. (Nakagami-m and Weibull distributions belong to different classes of distributions, while Rayleigh and Weibull belong to same class of distributions.) A homogeneous compound channel model Weibull × Weibull (WW) is also considered for reference and comparison. Appropriate performance metrics required for a transparent relay system are derived for the given compound channel models.
The rest of the paper is organized as follows. A brief discussion on the methodology adopted for analysis, metrics chosen for evaluation, and contributions are presented in the rest of this introduction. In Section 2, closely related work from the literature are discussed. In Section 3, the cooperative communication system and the channel model under consideration are presented. Exact expressions for the primary statistics like pdf and cdf are derived in Section 4. Performance measures such as coefficient of variation, average SNR, and outage probability are computed in Section 5. Using simulation studies, validation of analytical results, determination of average BER performance of a transmission system, and study of quantile-quantile (q-q) profiles for different channel models are done in Section 6. Discussion on the results are included in Section 7, and the paper is concluded in Section 8.
1.1 Summary of methodology, metrics, and contributions
Instead of exploring new distributions for the end-to-end compound channels, a closer match to the actual scenario can be obtained if product combinations of known distributions are selected according to the topographical variations. This results in distributions that show accurate fit to the practical scenario. Analytical study of this unexplored framework for the compound channels in multi-hop AF relay communication is too complex and often uses approximate solutions. The generic approach that we propose can be used to obtain the exact probability density functions (pdfs) of new random variables (RVs) that are algebraic combinations or functions of independent RVs.
Moment generating function (mgf) and Mellin transform (MT) are the two commonly used tools for the computation of distribution moments [6, 7] in communication theory. As we focus on a generic approach for finding the pdf of a heterogeneous compound distribution, the product convolution property of MT for independent RVs is exploited. Also, as MT translates exponentials to polynomials, complexity of the analysis can thus be reduced to a certain extent. Using certain other properties of MT, convenient computation of signal-to-noise ratio (SNR) statistics is also possible. The ease of computation of metrics related to SNR, convenient extension to three or more hop counts, and computation of the end-to-end moments without the knowledge of overall pdf are some of the added benefits of this transform.
We can find the pdf from the inverse transform. The inverse transform computation requires the evaluation of a contour integral. For the case of known standard homogeneous compound distributions, this integral evaluation can be done directly by using the Meijer G-function which is available as built-in functions in computing software packages . But it is not readily available for every product distributions of interest. Therefore, conversion of the contour integral to appropriate Meijer-G format is essential for the evaluation. However, the coefficients of the transform variable may not be identical for two different distributions. This issue is solved in our work by applying gamma duplication formula and reduction rule to one of the component distributions.
Modeling the distribution of the signal fluctuation is just the first step. In order to employ a fading model in link budget, it is also vital to determine suitable performance metrics based on the statistical properties of the model. A performance metric involving higher order moments is often required to asses the possible signal variations. Hence, a metric called coefficient of variation (CV) is derived which reflects the slightest change in distribution with respect to the fading factors and hop counts. Performance prediction adaptive to hop count and terrain is essential for robust system implementation. Hence, outage probability is evaluated for different channel models using the exact pdf of SNR distributions.
As power allocation in AF system critically depends on the SNR and power constraints of relays, accurate estimation of average SNR, which is sensitive to hop count and individual hop distributions, is essential for the effective implementation of the system. Hence, an expression for the average SNR is derived (in terms of relay amplification factor). It is then used to simulate a two-hop AF relay transmission system and the bit error performance is analysed for different compound channel models. Metrics like CV and average SNR derived can be used to select appropriate branches in a selection or switched diversity combining systems [9, 10].
2 Related works
Most of the previous works on product pdf are based on mgf. When known standard distributions with the available mgf are used for the pdf and other statistics, readily available results from computing software packages can be used for evaluation.
In some of the previous works [6, 8, 11], harmonic distribution is assumed for the end-to-end SNR. But this is based on the condition that the real-time channel state information (CSI) is available at the relay node so that the relay amplification factor can be chosen accordingly. However, real-time CSI at relay node is impractical for the case of a transparent relay networks. In such situations, the end-to-end SNR distribution is to be obtained from the product distribution of the compound channel coefficients as we show in Section 3. Based on the upper bound obtained using the inequality between harmonic and geometric means of positive RVs, the moments of the end-to-end SNR for Rayleigh, Nakagami-m, and Nakagami-n fading channels are derived in ,. The approximation of harmonic distribution is valid only at high SNR, as per the derivation. Same approach based on bounds is followed in [11, 12] for Nakagami-m and generalized Gamma channels. The related works with non-identical constituent distributions either deal with same class of distributions with non-identical scale parameters or resort to approximations.
Exact primary statistics, pdf and cdf, for the product of n-Rayleigh RVs are derived in , using the Meijer G-function. The corresponding infinite series expansions of the expressions from subroutine packages are also provided. Exact expressions for capacity and SNR for n-Weibull distributions are derived in , where approximations to pdf are also provided. Using inverse Laplace transform of the mgf of product RVs, compound pdf for Nakagami-m distribution is derived in  in terms of Hyper geometric functions. Performance measures for a homogeneous n-Weibull distribution are derived in , where only the statistics based on moments of homogeneous compound channels are considered. The primary statistics like fading distributions and SNR distribution for homogeneous or heterogeneous compound channels are not obtained in this work.
In  and , Nakagami-m distribution is assumed for the individual channels, whose SNR distribution falls under another known standard distribution, namely Gamma. Since known identicalpdf s are used, for the constituent channels, mathematical subroutines can directly provide the corresponding pdf of the compound channel in special function formats. Using the Fox’s H-function, the pdf, cdf, and mgf for the N-product generalized Nakagami-m distribution for different m are found in , along with channel capacity and amount of fade (AoF). However, series form expansions of Fox’s H-function are tedious to obtain, especially when multiple order poles are involved. Using mgf approach, approximate average bit error rate and AoF are computed for identical n-Weibull distributions in . In this work, the end-to-end SNR is taken as the product distribution of individual SNRs for the AoF computation. The upper bound of SNR in a dual-hop system having non-identical Weibull distributions is considered in , where the overall SNR effect is approximated by selecting a channel having the minimum SNR. The same approach is followed in , for cascaded Rayleigh and Rican distribution, where effectively, the compound channel reduces to single channel model for further analysis. Despite these contributions, a unified approach for the exact characterisation of compound channels with combinations of distinct component distributions is not available in the literature.
An explicit investigation on the fading effects due to heterogeneous cascading and corresponding metrics evaluation is essential. Certain distributions, considered as special cases of other distributions, can be derived from the original distribution by parameter substitution. But for the integral inversion, when product effect is considered, parameter substitution to the original distribution may not produce accurate results. As the integral will only pick up the values of the residues at each pole, a single contour of integration is required for the inverse computation. This may not be always possible due to the change in the transform variable coefficients of the corresponding function arguments. This issue has to be addressed in order to analyse heterogeneous case, and it is in this aspect that our work is different from the above works.
3 Compound channel and system model
3.1 Compound channel model
is selected along with the Weibull distribution. The parameter m is also known as the fading factor and Ω=E[Y2] is the scaling factor. E[.] is the expectation operator. In Nakagami-m distribution, the exponentiated squared RV value ′y′ is multiplied by the fading factor (m). Hence, variability of this distribution will be less compared to Weibull distribution.
where λ2/2 denotes the variance. Note that the exponentiated RV in this case is raised to a constant power of 2 and belongs to the same class of Weibull distribution. Two cases of heterogeneous compound channels formed by the combinations of these distributions are considered to check the validity of the unified approach.
3.2 System model
where A j denotes the amplification factor of the jth relay, X i denotes the fading coefficient of ith hop, e l denotes the additive noise at the lth relay, and e n denotes the additive noise at the destination. As exact CSI is not available at intermediate relay nodes for a non-regenerative system, minimum amplification by the relay is considered as the bound (worst case condition). This imposes the condition that A1=A i =An-1=1, and therefore, effective channel coefficient v ef =V (defined in Equation 1). Under this situation, the relays act just like nodes that establish connectivity to the succeeding nodes. Moreover, considering normalised channel coefficients X i s, the additive noise effect get reduced due to channel multiplication. In short, the signal (data) distortion due to channel attenuation (multiplication) becomes much more significant than the additive noise.
4 Exact primary statistics of compound channels
In this section, the pdf and cdf of compound channels are derived, and the multiplicative effects of channel coefficients are investigated for homogeneous as well as heterogeneous cases. The approach can be extended to any combinations of distributions like Erlang, Beta, Gamma, etc., for which the MT are defined.
4.1 Probability density function
and it is denoted by M X (s), where s=a+j b∈C is a complex transform variable. This transform operator represents the second kind characteristic function. The usual techniques of conditioning on RV s or Jacobian transformations are no longer required if Mellin transform properties are applied; and the transform of the productpdf can be readily converted to moments of the compound channel pdf.
4.1.1 pdf of n-Weibull channel
See Appendix for the definition of general Meijer G-function.
4.1.2 pdf of heterogeneous channel - Nakagami × Weibull
where , , and . This is computed for specific values of m,β and is given in the Appendix.
4.1.3 pdf of heterogeneous channel - Rayleigh × Weibull
4.2 Cumulative distribution function
Note that the pdf expressions given in Equations 11, 17, and 23 have two variables, the transform variable s′ and the pdf argument v. Integrating the pdf with respect to the variable v to an integration limit of a given threshold value yields the exact cdf. The double integral can then be rewritten to express the inner integral in terms of v and the outer integral in terms of s′. The former one is then subjected to some algebraic manipulations (by making use of Gamma reduction rule) so that the exact cdf can be expressed in terms of Meijer G-functions.
4.2.1 cdf of the n-Weibull compound channel
4.2.2 cdf of Nakagami-m× Weibull channel
In order to plot the cdf, the function may be computed for some specific parameter values, as given in the Appendix.
4.2.3 cdf of Rayleigh × Weibull channel
5 Application of the results-performance evaluation metrics
As suggested in , in the general context of describing the behaviour of a diversity system with arbitrary combining techniques, the channel quality based on channel statistics can be used for performance evaluation. Three different measures have been investigated to evaluate the performance of multi-hop networks using diversity schemes.
5.1 Coefficient of variation-CV
5.2 Average SNR
Under the assumption of constant that averages to unity in each single hop, becomes A2, where
Various moments of γ are computed from this by setting s=p+1. Using this expression for the average SNR computation, and by selecting scaling factors proportional to path-loss due to distance, the appropriate hop distance and relay amplification factor can be designed for a threshold average SNR for each compound distribution.
5.2.1 Average SNR of n-Weibull channel
5.2.2 Average SNR of NW channel
5.2.3 Average SNR of RW channel
Comparison of Equations 46 and 40 reveals that the average SNR of RW compound channel reduces to that of a single-hop Weibull channel if λ is unity, since Γ(2)=1, and λ.α=α2. (With reference to Equation 4, λ=1 corresponds to Rayleigh distribution with variance 0.5.)
A single-stream two-hop AF relay transmission system was simulated for a relay amplification factor of 2, to investigate the error performance of the system over each compound channel.
5.3 Outage probability
which is thecdf of γ, evaluated at γ=γth. To compute the outage, the pdf of SNR is to be obtained first through the inverse MT approach.
5.3.1 Outage probability for n-Weibull channel
5.3.2 Outage probability for NW channel
5.3.3 Outage probability for RW channel
6 Simulation studies
where . The Rayleigh channel coefficients were generated in the same way with β=2, and the Nakagami-m using the Gamma random variates from the Matlab software package. Using these random variates generated for the different constituent channel coefficients with various parameter values, the corresponding product channel coefficients were obtained for different hop count values.
Comparison of analytical cdf to simulation cdf
Threshold ( v t )
where represents an indicator function which takes value unity when v i ≤v t , and Z represents the total number of observations.
The empirical CV, denoted as CV e , was computed from the samples generated for various values of β,m and n. For the computation of the sample moments and CV e , 106 samples were generated in each of the homogeneous and heterogeneous compound channel cases. The exact match between the theoretical CV and Empirical CV shows the accuracy of the exact expressions derived for CV and moments.
7 Results and discussions
The analytical and simulation results obtained are discussed in this section. The plots of derived analytical pdf for Weibull × Weibull(WW), Nakagami × Weibull (NW), and Rayleigh × Weibull (RW) are shown in Figure 2. It may be noted that the pdf of WW distribution approaches Gaussian distribution for β=4. These exact pdf expressions can be used to identify the distribution from empirical data, using q-q plots.
Figure 3 shows the plots corresponding to the derived exact expressions of cdf for different values of n and for the NW, RW cases. Knowledge of cdf can be used to fix the dynamic range (DR) at the receiver unit.
The DR can be obtained as the ratio vt max/vt min and the thresholds are decided based on the cdf values. It is seen from Figure 3 that the required DR at receiver will have to be increased as the order of multiplication increases. The smaller slope of cdf for RW and NW cases, compared to the 2-Weibull (WW) case, and the corresponding difference in DR are obvious from Figure 3. Simulation results show exact match to the analytical results as tabulated in Table 1.
Lower values of CV indicate better regularity of the channel coefficients. In Figure 4, CV is plotted against fading factors β or m. For β=2, the RW and WW distributions give the same CV, as both distributions reduce to RR (two-Rayleigh) distribution. Among the three distributions, NW shows the minimum variability (most regular), as expected. Simulation and analytical values (CV e , CV) show exact match in Figure 4, proving the accuracy of the derived analytical expressions. The exact match between the theoretical CV and Empirical CV shows not only the accuracy of the expressions derived for CV but also the exact agreement of the derived (population) moments to the sample moments.
The average SNR computed analytically and through simulation (as discussed in Section 6) for each case are plotted against the relay amplification factor in Figure 5. The close match between the simulation and theoretical plots observed in Figure 5 shows the accuracy of the derived expressions for average SNR of WW, RW, and NW channel models. The lowest average SNR for the NW channel under the given fading factors (β=m=1) means that the required amplification factor for a target average SNR for a system operating over this channel will be higher compared to that of RW or NW channels. Based on Figure 5, the required overall amplification factor for any target average SNR can be computed.
The plot of 1-Pout in Figure 6 gives the probability of the received SNR to be above a threshold, for n-Weibull compound channel with various hop counts. Based on the minimum γth required at the receiver, the number of hop counts can be selected. Also shown in Figure 6 are the plots for the NW and RW dual-hop compound channels. The improved performance of RW case over 2-Weibull case, in the high SNR region, and the degradation of NW channel over the homogeneous case are exposed in Figure 6.
In order to demonstrate the application of the derived average SNR, a two-hop transparent relay transmission system was simulated as discussed in Section 6. The average BER performance of the system over NW, RW, and WW channels are plotted in Figure 7. Obviously, maximum error rate occurs with NW channel as it delivers minimum average SNR. For the system operating over NW channel, an additional 3-dB increment in is required for achieving an average BER of 10-3 in comparison with WW channel scenario.
From the derived exact expressions for the various primary statistics and the corresponding simulation results, it is evident that the end-to-end channel distribution for different number of hops may deviate significantly in modality, skewness, Kurtosis, etc., due to cascading. The derived performance metrics depend on the number of hops and the constituent distributions. These exact expressions, and the unified approach that can be used for other cases as well, are expected to be of great use in the realistic system design and performance evaluation of the multi-hop, diversity attaining transparent cooperative systems.
Exact expressions are derived for the moments, pdf, and cdf, of the proposed heterogeneous compound channel models in multi-hop transparent cooperative relay transmissions for outdoor applications. The statistical properties of heterogeneous compound channels, Nakagami × Weibull and Rayleigh × Weibull, are compared with that of homogeneous compound channel assumption. The expressions are derived in tractable compact forms, in terms of Meijer G-functions, so that accurate and easy computations are possible. Coefficient of variation, average SNR, and outage probability are the performance metrics derived to quantify the variability, fidelity, and acceptable SNR level of the system, respectively. Corresponding to the SNR statistics, a two-hop transparent relay transmission system is simulated, and the average BER performance of the system under the proposed channel models are also evaluated. The approach can be easily extended to other combinations of distributions also, if the constituent channels are Mellin transform defined. The results are expected to be useful in the design of receivers in multi-hop diversity attaining cooperative relay systems, to achieve predefined QoS.
A.1 General formulae used
where Γ(.) is the complete gamma function  [Eqn.9.301].
Gamma duplication formula
Gamma reduction rule
A.2 Analytical Steps for the Integral Evaluations
The inverse integral equations need to be reduced into a computable Meijer G-function format. As software packages do not have built-in subroutines for the evaluation of such integrals for heterogeneous combinations of distributions, appropriate format conversion is necessary. For illustration, details of the conversion of pdf of NW channel and cdf of RW channel are given below.
A.2.1 Steps for the computation of pdf for NW case
A.2.2 Steps for the computation of cdf for RW case
Adopting similar steps for the integral equations of SNR will facilitate the computations of SNR statistics also.
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