Outage probability of fixed-gain dual-hop relay selection channels with heterogeneous fading
- Yue Wang^{1} and
- Justin P Coon^{2}Email author
https://doi.org/10.1186/s13638-015-0424-0
© Wang and Coon 2015
Received: 16 January 2015
Accepted: 6 July 2015
Published: 23 July 2015
Abstract
In this paper, we develop a framework that can be used to analyze the outage probability of a dual-hop fixed-gain amplify-and-forward relay system performing single relay selection. We consider three selection strategies: (1) first-hop selection, whereby the relay is chosen to maximize the signal-to-noise ratio (SNR) at the relay irrespective of the second-hop channels; (2) second-hop selection, in which the relay is chosen such that the second-hop SNR is maximized; and (3) dual-hop selection, where the end-to-end SNR is maximized. The proposed analytical framework is capable of treating systems operating in heterogeneous fading conditions, where all or a subset of the source-relay and relay-destination channels has non-identical distributions or even experience completely different fading processes. We apply the framework to calculate new exact series representations of the outage probability for the three aforementioned cases of relay selection when all channels adhere to the Nakagami-m model. Our analysis is corroborated by simulations. Finally, we provide a discussion of how the proposed framework can be applied to analyze other fading configurations using the techniques detailed herein.
Keywords
1 Introduction
The paradigm of cooperative communication has been shown to provide significant advantages in cellular networks (and many other applications) for more than a decade [1]. With the deployment of LTE networks across the world in recent years [2] and the shifting of focus in the research community to 5G cellular technology [3], the ideas surrounding cooperative communication are evolving. Of particular interest in the discussion of 5G is relay-aided communication. The heterogeneous structure of 5G networks will consist of layers of macrocells, small cells, relays, and device-to-device (D2D) networks [4]. In areas lacking a wired backhaul infrastructure, relays are, and will continue to be, deployed to act as picocell base stations to local user equipments (UEs) and to mimic (for the most part) a UE in the view of the macrocell base station. Within the D2D networking paradigm, relays will form a key component in the control infrastructure of a 5G network, linking D2D networks to the macrocell base station [4]. This idea can be further extended to connect D2D clusters to the macrocell by using one of the constituent UEs in the cluster as a relay [5, 6].
Fixed-gain amplify-and-forward (FGAF) relay systems have attracted a lot of attention recently due to their low complexity in practical implementation. To date, numerous efforts have been devoted to the performance analysis of dual-hop FGAF relay systems (see [7] and the references therein), including error rate, capacity, and outage calculations. With regard to the latter, closed-form outage probability expressions for dual-hop FGAF were derived in [8] for Rayleigh fading channels, in [7, 9, 10] for Nakagami-m channels, and in [11] for Rician channels. In [12], a moment generation function (MGF) approach was used to analyze the performance of dual-hop FGAF systems operating in arbitrary, or generalized, fading conditions. Heterogeneous fading conditions were also considered in [13] in the context of systems employing maximum ratio combining and transmit antenna selection at the relay. Many of these reported results were recently generalized and incorporated into a multi-hop framework in [14].
- 1.
Much of the published literature considers FHS or DHS, neglecting the case of SHS despite the fact that the latter is an important practical model.
- 2.
Source-relay (S-R) and relay-destination (R-D) channels have largely been assumed to follow Rayleigh, Nakagami-m, or Rician distributions in relay selection systems, with some important fading distributions having been largely ignored (e.g., Weibull and Hoyt).
- 3.
Heterogeneous fading conditions have generally not been studied for relay selection channels, although such scenarios may arise in a plethora of applications, such as a cellular base station transmitting to a relay located in an enterprise zone, which then conveys the signal to users inside an office block.
In this paper, we aim to overcome the aforementioned limitations by presenting a framework for analyzing the outage probability of FGAF systems employing relay selection in heterogeneous channel conditions. The proposed framework applies generally for arbitrary configurations of S-R/R-D channels, e.g., Nakagami-m/Rician, Hoyt/Weibull, etc. In order to facilitate exposition, we focus on Nakagami-m fading in this work and derive the outage probability for systems operating in heterogeneous channels for FHS, SHS, and DHS selection strategies. We also provide complete diversity results for each selection mechanism and qualitatively discuss the application of the framework to other fading configurations.
The rest of the paper is organized as follows. The dual-hop FGAF system model is presented in the “System model” section. The analytical framework is detailed in the “Analytical outage probability framework” section, which is then applied in the “Analysis for Nakagami-m ” section to study heterogeneous Nakagami-m scenarios. The analysis is corroborated with numerical results in the “Simulation results and discussion” section. Extending the framework to other more general fading configurations is discussed and concluding remarks are given in the “Conclusions” section.
2 System model
Define \(\bar {\gamma }\) as a reference average received SNR parameter. Clearly, the average received SNR values corresponding to different hops will be different in practice. This is accounted for in the model by denoting the variances of the zero mean additive white Gaussian noise at the nth relay and the destination as \(\sigma ^{2}_{1,n}\) and \({\sigma ^{2}_{2}}\), respectively, and defining parameters ρ _{1,n } and ρ _{2} such that \(\bar \gamma = \rho _{1,n}/\sigma ^{2}_{1,n}\) for n=1,…,N and \(\bar \gamma = \rho _{2}/{\sigma ^{2}_{2}}\). Thus, the SNR at the nth relay of the first hop is given by \(X_{n}=\frac {|h_{1,n}|^{2}}{\rho _{1,n}}\bar {\gamma }\), and the SNR corresponding to the local channel at the destination (i.e., not the end-to-end SNR) is \(Y_{n}=\frac {|h_{2,n}|^{2}}{\rho _{2}}\bar {\gamma }\).
For the relay selection process, a single relay is chosen among the N available relays according to the FHS, SHS, or DHS selection criteria. For FHS, the selection is made at the source to maximize the received SNR at the relay. The chosen relay can be notified of its selection through, for example, a dedicated control channel. Exact details of how control signaling might be performed are beyond the scope of this paper. Consequently, under the FHS approach, the kth relay is selected such that k= arg max{X _{ n }}. Analogously, for SHS, we have k= arg max{Y _{ n }}. Finally, for DHS, global channel knowledge is used to select the relay that maximizes the end-to-end SNR, which we denote by γ _{ n } for the nth path. Thus, in this case, we can write k= arg max{γ _{ n }}.
3 Analytical outage probability framework
Note that we have slightly abused notation here since P(γ _{ k }<z) is conditioned on the selection mechanism (FHS or SHS) and the index of the selected relay k. Nevertheless, the important point is that γ _{ k } is a function of two independent variates, X _{ k } and Y _{ k }. This enables us to write a general formula for the cumulative distribution function (c.d.f.) of γ _{ k }, which we encompass in the following lemma (the proof is given in the Appendix).
Lemma 1.
Although the lemma seems somewhat inaccessible due to the integrals, we note that it provides an exact expression as a functional of the first- and second-hop fading density functions, regardless of the underlying distributions. The only stipulation is that the channels corresponding to the two hops are statistically independent, a condition that is often, if not always, satisfied in practice. We will revisit that and apply this lemma to a great effect in the next section.
where it is assumed that {X _{ n }} are statistically independent. The expression for ℘ _{SHS,k } is the same, but with Y replacing X.
Since λ _{ k }(s,z) is determined by the channel distributions, it can often be calculated easily. For the contour integral, we will exploit the pole structure of λ _{ k }(s,z) and apply the residue theorem from complex analysis to solve the integral, thus yielding series expressions of the outage probabilities in general. These technical details will be the focus of the next section.
4 Analysis for Nakagami-m
For the purpose of presenting how the analytical framework can be applied, we assume the channels for the two hops adhere to the same class of distribution, but may have different parameters. The same approach can be used to study arbitrary combinations of fading distributions for the different channels.
where \(\gamma (a,x) = {\int _{0}^{x}}{t^{a-1}e^{-t} \,\mathrm {d} t}\) is the lower incomplete gamma function [22]. The parameter m _{1,n }>1/2 is the shape parameter of the distribution (denoted by m _{2,n } for the p.d.f. of Y _{ n }) while \(\theta _{1,n} = \mathbb E[|h_{1,n}|^{2}]\) is effectively the scale parameter (correspondingly, θ _{2,n } for Y _{ n }).
4.1 FHS
where the summation is (N−1)-fold, with the indices being the set \(\mathcal L_{k} = \{l_{1},\ldots, l_{k-1}, l_{k+1}, \ldots, l_{N}\}\), with l _{ n }=0,1,2,… for all n. In addition, \(\beta _{1}=\sum _{n=1}^{N}{\rho _{1,n}}/{\theta _{1,n}}\) and \(\tilde {m}_{1,k}=\sum _{n=1}^{N} m_{1,n}+\sum _{n=1,\,n\neq k}^{N}l_{n}\).
is the confluent hypergeometric function (Tricomi’s solution) [23].
Applying (16) to (8) gives the outage probability for FHS in Nakagami-m channels.
Although the outage probability is expressed as an infinite series, we will verify later that truncating the series to only a few terms provides an excellent approximation. Moreover, the hypergeometric function U(·,·,x) is well behaved for small arguments^{6}, and thus, it is possible to construct an asymptotic series of Poincaré type for \(\bar {\gamma } \rightarrow \infty \). Discussion of such an activity is rather involved and would detract from the focus of this paper; hence, we defer it to a future contribution.
We also note that the outage expression given above applies to the general case where each of the N available channels at a given hop is subject to non-identical Nakagami-m fading with different scale and shape parameters, i.e., m _{1,n }≠m _{1,q } for n≠q and so on. The result is therefore more general than that given in [19], where the outage probabilities are obtained for the case where m _{1,n }=m _{1} and m _{2,n }=m _{2} for all n.
To conclude this subsection, we provide a brief note on the case where m _{2,k } is an integer. Under this condition, λ _{ k }(s,z) has second-order poles at s=−m _{2,k },−m _{2,k }−1,…. The residue theorem can still be applied, but the calculations become cumbersome, and careful bookkeeping must be enforced in order to ensure the resulting series expansion is accurate. This is particularly true when \(\tilde {m}_{1,n}\) has certain properties (such as being an integer), as this will cause the structure of the expansion to vary. For our practical purposes, however, it is much more intuitive and efficient to evaluate the outage probability using non-integer values of m _{2,k } to approximate integers. We will show through simulations that this simple approximation works very well in practice.
4.2 SHS
where β _{2} and \(\tilde {m}_{2,k}\) are defined in a similar manner to β _{1} and \(\tilde {m}_{1,k}\).
4.3 Single relay
where θ _{ q } and m _{ q } are the scale and shape parameters for the Nakagami-m channel at the qth hop, and \(\rho _{q}={\sigma ^{2}_{q}}\bar {\gamma }\) where \({\sigma ^{2}_{1}}\) and \({\sigma ^{2}_{2}}\) are the noise variances at the relay and the destination, respectively.
Although an outage expression for dual-hop FGAF systems with a single relay operating in Nakagami-m fading channels was provided in [7, eq. (20)], the result presented here is simpler and easier to evaluate numerically. To the best of our knowledge, (18) has not been reported in the literature.
4.4 DHS
The outage probability for DHS is obtained by substituting the expression for P _{o}(z) given by (18) for the quantity in the brackets of (7), i.e., \(1-\frac {1}{2\pi i}\int _{c-i\infty }^{c+i\infty }\lambda _{k}(s,z) = P_{\mathrm {o}}(z)\). In doing so, all parameters should be replaced by those corresponding to the kth path for k=1,…,N. We omit the explicit expression for brevity.
4.5 Diversity analysis
Diversity orders for FHS, SHS, and DHS in Nakagami-m fading
Selection scheme | Diversity order |
---|---|
Single relay | min{m _{1},m _{2}} |
FHS | \(\min \left \{\sum _{n=1}^{N} m_{1,n}, m_{2,1}, \ldots, m_{2,N}\right \}\) |
SHS | \(\min \left \{\sum _{n=1}^{N} m_{2,n}, m_{1,1}, \ldots, m_{1,N}\right \}\) |
DHS | \(\sum _{n=1}^{N}\min \{m_{1,n}, m_{2,n}\}\) |
5 Simulation results and discussion
In this section, we present numerical results in an effort to validate the analysis presented above. For all of the results presented below, we consider the case where there are two relays available, i.e., N=2, and the ρ and θ parameters are all set to 1 if not otherwise stated.
Shape parameter definitions for different dual-hop Nakagami-m fading configurations. Values that span two columns indicate a single-relay system
Configuration | m _{1,1} | m _{1,2} | m _{2,1} | m _{2,2} |
---|---|---|---|---|
Nak-1 | 1.3 | 2.5 | ||
Nak-2 | 4 | 4 | ||
Nak-3 | 1.3 | 0.7 | 2.5 | 2.5 |
Nak-4 | 2.5 | 2.5 | 1.3 | 0.6 |
Nak-5 | 2.5 | 2.5 | 2.5 | 2.5 |
Nak-6 | 0.9 | 1.3 | 2.5 | 3.5 |
Diversity orders corresponding to configurations outlined in Table 2
Configuration | FHS | SHS | DHS |
---|---|---|---|
Nak-1 | 1.3 | 1.3 | 1.3 |
Nak-2 | 4 | 4 | 4 |
Nak-3 | 2 | 0.7 | 3.2 |
Nak-4 | 0.6 | 1.8 | 3.1 |
Nak-5 | 2.5 | 2.5 | 5 |
Nak-6 | 2.4 | 0.9 | 3.4 |
6 Conclusions
The framework presented herein can be applied to analyze many practical system configurations. The key components that are required are derived from Lemma 1, whence we see that density functions of the effective channel gains X _{ k } and Y _{ k } are required along with integrals that are closely linked to their Mellin transforms. Certainly, having this information available makes the calculation of ℘ _{ h,k } possible in re (6) as a series expansion if not in a simple closed form.
for some shape and scale parameters m and θ, the poles of which lie at s=−m(1+j) for j=0,1,…. Thus, we can see that problems involving Weibull distributions will also experience a rich pole structure, and the full power of the residue theorem will be applicable.
Other fading distributions are also worth considering within this framework. Rician and Hoyt densities involve Bessel functions, which lead to hypergeometric functions (_{1} F _{1} in the case of Rician and _{2} F _{1} in the case of Hoyt) through the evaluation of the integral \(\int x^{s} f_{X}(x) \,\mathrm {d} x\) in (5). But these functions are either entire or meromorphic in the arguments and thus can be dealt with by calculating the residues just as was done for Nakagami-m fading. Although the method is straightforward, the exact results are often challenging to interpret in written form given only a few pages in which to do so, and thus, we have omitted them from this discussion.
It should be clear by now that the fading distributions corresponding to each hop in the network need not belong to the same class, e.g., Rician, Weibull, etc. Indeed, the power of the framework is that it is relatively agnostic to the functions f _{ X } and f _{ Y }, as long as the related integrals given in (5) exist and the result facilitates contour integration vis-à-vis (4).
As a final note, we have liberally made use of infinite series representations of special functions through this analysis. For the examples discussed in the previous section, these series were convergent and thus well behaved under the operation of integration. However, if one is concerned mostly with asymptotics (e.g., high SNR), this condition need not be satisfied. In fact, very accurate approximations to the outage probability of some relatively complex systems can be attained from truncated asymptotic series. Again, we leave the details to another forum due to space restrictions.
7 Endnotes
^{1} The term partial selection is also used in the literature to describe relay selection based on channel knowledge pertaining to a single hop.
^{2} As is customary, we consider SNR outage since there is a one-to-one relationship between this performance metric, mutual information outage, and many symbol-error-rate outage expressions encountered in practice when flat fading is assumed.
^{3} This constant arises from the use of the Mellin inversion theorem [25] and basically separates the poles of the integrand in a particular and well-defined way for a given function λ _{ k }. The interval in which c lies is explored for specific examples in the next section.
^{4} This follows from Stirling’s formula [22] and an application of Jordan’s lemma [26].
^{5} We will return to the case where m _{2,k } is an integer later.
^{6} Technically, there is a singularity at x=0, but for the purposes of our system analysis, this condition will never hold.
^{7} When the shape parameters are not distinct, the same approach that is outlined in this section can be taken, but the algebra and differentiation become cumbersome.
8 Appendix
for probability distributions with exponentially decaying tails. Finally, we recall that given two independent random variables X and Y and their product Z=X Y, the Mellin transform of the density f _{ Z } is the product of the transforms of the densities f _{ X } and f _{ Y }, i.e., \(\mathbb M[f_{Z};s] = \mathbb M[f_{X};s] \mathbb M[f_{Y};s]\).
The result stated in the lemma follows by using (31) along with the inversion formula (30) and substituting into (32).
Declarations
Acknowledgements
Part of this work was undertaken while the authors worked at Toshiba Research Europe’s Telecommunications Research Laboratory in Bristol, UK.
Authors’ Affiliations
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