Performance analysis of dualhop satellite relay networks with hardware impairments and cochannel interference
 Guo Kefeng^{1}Email authorView ORCID ID profile,
 Guo Daoxing^{1},
 Huang Yuzhen^{1} and
 Zhang Bangning^{1}
https://doi.org/10.1186/s1363801709063
© The Author(s) 2017
Received: 9 May 2016
Accepted: 26 June 2017
Published: 17 July 2017
Abstract
In this paper, we investigate the impact of hardware impairments on the outage performance and the instantaneous capacity of dualhop decodeandforward (DF) satellite relay networks in the presence of cochannel interference. Specifically, the exact closedform expression for the outage probability and the analytical expression for the instantaneous capacity of the system are derived over a shadowedRician fading channel, which provides a fast means to evaluate the impact of hardware impairments on the system performance. Our results demonstrate that the outage floor occurs and the capacity upper bound appears when the hardware impairments exist. In addition, simulation results were provided to verify the correctness of the analytical results.
Keywords
Satellite relay network Hardware impairments Cochannel interference Outage probability Instantaneous capacity1 Introduction
A relay is used to improve the coverage, reliability, and quality of service in wireless systems, which has become a hot research topic over the past decade [1–4]. A satellite is used as the relay to help a source transmit a signal in a farther distance when the source itself cannot transmit directly. Satellite relay communication is capable of providing seamless connectivity and highspeed broadband access for worldwide users, especially in the fields of broadcasting, disaster relief, and navigation, where the deployment of wired and wireless terrestrial networks is not economically applicable.
The links between the ground and a satellite are usually modeled by composite fading distribution to describe more accurately the amplitude fluctuation (large scale) of the transmitted signal’s envelope. To this end, the shadowedRician distribution has been proposed in [5]. It approaches the lineofsight (LOS) communication using the Rician distribution, while the fluctuation of the signal’s amplitude is Nakagamim distributed. Hence, it agrees very well with experimental data, whereas it demonstrates quite a manageable mathematical tractability.
Capitalizing on the latter features of the shadowedRician distribution, an analytical performance evaluation of dualhop amplifyandforward (AF) systems with a satellite relay, was provided in [6] and [7]. In [8] and [9], a singleantenna land mobile satellite (LMS) system was investigated, where the source is a satellite and the relay/destination are terrestrial nodes. Furthermore, a similar LMS system with the sourcetorelay satellite link was investigated in [10].
However, most research in the area of satellite relaying assumes that the transceiver hardware of the network node is perfect. This cannot be achieved in practical communication systems. In practice, hardware suffers from several types of impairments, e.g., phase noise, I/Q imbalance, and high power amplifier (HPA) nonlinearities [11–13]. The impact of hardware impairments on singlehop wireless communication was broadly investigated in [14–16]. In addition, as indicated in [17–20], there is a fundamental capacity ceiling due to the hardware impairments, which cannot be improved by increasing the transmission power. In [17], the authors analyzed the performance of multipleinputmultipleoutput (MIMO) system; the capacity and the outage probability of the system were derived. In [18], the authors analyzed the system performance of the fullduplex relaying network and found that the system capacity had an upper bound in the presence of hardware impairments. In [19], the opportunistic and partial relaying selection was used in which a new way was derived to obtain the closedform expression for the system performance. In [20], the authors analyzed the impact of hardware impairments on the fixedgain AF relaying Nakagamim fading channels. Above all, the system channel they analyzed was just a Rayleigh or Nakagamim channel; there are scarcely any papers that analyzed the effect of hardware impairments on the shadowedRician channel.
Although the aforementioned works have significantly improved our understanding on the performance and benefits of a satellite relay network, they only considered a noiselimited scenario and the ideal case of no cochannel interference (CCI) at the destination, which are unrealistic due to the current aggressive reuse of spectrum resources. In recent years, a great deal of research has focused on the performance of conventional terrestrial systems in the presence of CCI over Rayleigh or Nakagamim fading channels [21–23]. Due to the reuse of spectrum resources in practice, the major challenge facing satelliteterrestrial networks is the problem of interference from intercomponent and/or intra component networks [24]. The authors in [25] first investigated the performance of satelliteterrestrial networks with CCI at the destination for the single antenna scenario and analyzed the system performance.
As described above, we find that it is significant to analyze the impact of hardware impairments and CCI on the satellite relay communication network in shadowedRician channel. Until now, to the best of the authors’ knowledge, there is scarcely any paper that analyzes the impact of hardware impairments on a satellite relay network with CCI. To fill this gap, we herein investigate the joint impact of hardware impairments and CCI on the performance of a dualhop satellite relay network.

Firstly, the closedform expression for the outage probability is derived which is applicable to cases involving an arbitrary number of interferers, antennas, and hardware impairments, and hence, the derived expression provides a fast means to investigate the system performance.

Secondly, the analytical expression for the instantaneous capacity of the system, which can be used to efficiently evaluate the instantaneous capacity at high signaltonoiserates (SNRs), is derived.

Thirdly, in this paper, the system performance is analyzed under two channel conditions, that is, channel parameters are integers or arbitrary values.
The rest of this paper is organized as follows. In Section 2, the model we analyzed is briefly introduced. In Section 3, the cumulative distribution function (CDF) of the endtoend signaltointerferenceplusnoiseanddistortionratio (SINDR), the exact expression for the outage probability, and the analytical expression for the instantaneous capacity are derived. In addition, some system performance metrics are provided. In Section 5, the analytical results are presented. Finally, in Section 5, a brief summary of our work for this paper is presented. In the Appendix, the detailed proofs for some Lemmas and Theorems are given.
2 System and channel models
where η _{1i } denotes the distortion noise with zero mean and variance \(k_{1i}^{2}P_{1i}\) and k _{1i } is sufficient to characterize the aggregate level of impairments of the ith antenna of S →R channel [26].
where h _{2} is the independent but nonidentically distributed shadowedRician RV between R and D and g _{ j } denotes the channel coefficient between the jth interference (I) and D which is an independent but nonidentically distributed Rayleigh RV. y _{ r } is the signal received from R with average power P _{2}=E[y _{ r }^{2}], s _{ j } is the signal from the jth interference with power P _{ j }=E[s _{ j }^{2}], v _{ d } is the AWGN which is represented as \(v_{d} \sim {\mathcal {C}}{\mathcal {N}}\left ({0,{\delta _{d}^{2}}} \right)\) at D, η _{2} denotes the distortion noise with zero mean and variance \(k_{2}^{2}P_{2}\), η _{ j } denotes the distortion noise with zero mean and variance \(k_{j}^{2}P_{j}\) of the I, and k _{2} and k _{ j } are sufficient to characterize the aggregate level of impairments of the R→D channel and I→D channel, respectively [26]. To simplify the analysis, we assume that the hardware impairment level of the I→D channel has the same value as \(\phantom {\dot {i}\!}k_{1}={\ldots }=k_{j}={\ldots }=k_{I_{1}},j=1,{\ldots },I_{1}\).
3 System performance
In this section, we present a comprehensive performance investigation of the system described in the previous section. Specifically, the description of the shadowedRician channel, the closedform expression of the outage performance, and the analytical expression of the instantaneous capacity of the considered system with hardware impairments are derived.
3.1 The endtoend SINDR of the system
where \({\lambda _{1i}} = \frac {{{{\left  {{h_{1i}}} \right }^{2}}{P_{1i}}}}{{\delta _{r}^{2}}}\) and \(k_{1i}^{2} = {\Delta _{1i}}\).
where \({\lambda _{2}} = \frac {{{{\left  {{h_{2}}} \right }^{2}}{P_{2}}}}{{\delta _{d}^{2}}}\), \({\lambda _{j}} = \frac {{{{\left  {{g_{j}}} \right }^{2}}{P_{j}}}}{{\delta _{d}^{2}}}\), \(k_{2}^{2} = {\Delta _{2}}\) and \(k_{j}^{2} = {\Delta _{j}}\).
3.2 The shadowedRician channel
where { l,p } ∈ { { 1,i},{2} }, \({{{\overline {\lambda }}_{lp}}}\) is the average SNR of each link, \({\alpha _{lp}}\mathop = \limits ^{\Delta } \frac {{{{\left ({\frac {{2{b_{lp}}{m_{lp}}}} {{2{b_{lp}}{m_{lp}} + {\Omega _{lp}}}}} \right)}^{{m_{lp}}}}}} {{2{b_{lp}}}}\), \({\beta _{lp}}\mathop = \limits ^{\Delta } \frac {1} {{2{b_{lp}}}}\), \({\delta _{lp}}\mathop = \limits ^{\Delta } \frac {{{\Omega _{lp}}}} {{2{b_{lp}}\left ({2{b_{lp}}{m_{lp}} + {\Omega _{lp}}} \right)}}\), Ω _{ lp }, 2b _{ lp } and m _{ lp }≥0 correspond to the average power of the LOS component, the average power of the multipath component, and the fading severity parameter ranging from 0 to ∞, respectively. In the following, we consider the two cases of m _{ lp } being either an integer value or an arbitrary value. In (7), _{1} F _{1}(a _{1};b _{1};z) denotes the confluent hypergeometric function defined in [27].
3.2.1 Integer m _{ lp } value case
where (·)_{ q } is the Pochhammer symbol.
3.2.2 Arbitrary m _{ lp } value case
3.3 The outage performance
where P _{ out }(x _{0}) is the outage probability of the system.
Next, \({F_{{\gamma _{1}}}}\left ({{x_{0}}} \right)\) and \({F_{{\gamma _{2}}}}\left ({{x_{0}}} \right)\) are derived respectively.
Proof
With the help of (4), (10), and (13), \({F_{{\gamma _{1}}}}\left ({{x_{0}}} \right)\) can be represented as (16), and (17) in both cases, which are shown at the top of this page and next page, respectively. □
Lemma 2
With the help of (5), \({F_{{\gamma _{2}}}}(x_{0})\) is given by (19) and (20), respectively.
Proof
With the help of (10), (13), and (18), the closedform expression for \({F_{{\gamma _{2}}}}(x_{0})\) is derived as (19) and (20) respectively; a detailed proof is given in the Appendix. □
Theorem 1
When m _{ lp } is an integer, \({F_{{\gamma _{e}}}}(x_{0})\) can be given by (21).
Proof
□
Theorem 2
When m _{ lp } is an arbitrary value, \({F_{{\gamma _{e}}}}(x_{0})\) can be given by (22).
Proof
□
3.4 Instantaneous capacity
In this section, we analyze the instantaneous capacity of the system, and the analytical expression for the instantaneous capacity of the system is given. The approximate value of the system at high SNRs region is also derived.
where γ _{ e } is the SINDR of the system and C is the instantaneous capacity of the system.
4 Numerical results
In this section, Monte Carlo simulations are provided to verify the analytical results. Some representative numerical results are provided to evaluate the impact of hardware impairments of the considered systems. Here, the analytical curves of arbitrary the m _{ lp } value case are obtained by using the first 30 terms of the infinite series. In this section, we assume that m _{2}=5,b _{2}=0.251,Ω _{2}=0.279 for the channel parameters between R and D, k _{ j }=0.1, P _{1i }=P _{ r } and \(\frac {{{P_{j}}}}{{\delta _{d}^{2}}} = 1dB\). We also assume that N _{1}=3 and I _{1}=3 for the simulations. As shown, the analytical results agree well with the Monte Carlo simulations, which imply the correctness of the derived theoretical formula.
5 Conclusions
In this paper, we investigated dualhop satellite relay networks with hardware impairments and cochannel interference. Specifically, the exact closedform expression for the outage probability of the considered system was derived, which made it easy to evaluate the impact of the impairing parameters on the system performance. The results revealed that an outage floor appeared when the hardware impairments existed. In addition, the analytical expression for the instantaneous capacity was also derived, which revealed that the instantaneous capacity had an upper bound when the system had hardware impairments. However, when the channel was under heavy fading, the performance of the system was worse.
6 Appendix
where \(A = diag\left ({{\mu _{1}},{\mu _{2}}, \cdots {{\mu _{{I_{1}}}}}} \right)\), ρ(A) is the number of district diagonal elements of A, μ _{〈1〉}>μ _{〈2〉}>⋯>μ _{〈ρ(A)〉} are the district diagonal elements in decreasing order, τ _{ i }(A) is the multiplicity of A, and χ _{ i,j }(A) is the (i,j)th characteristic coefficient of A [30].
With the help of [27. section 3.351.3], and after simplifications, (19) and (20) are presented.
Declarations
Acknowledgements
This work is supported by the National Science Foundation of China (No.61501507) and the Jiangsu Provincial Natural Science Foundation of China (No.20150719).
Competing interests
The authors declare that they have no competing interests.
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References
 JN Laneman, DNC Tse, GW Wornell, Cooperative diversity in wireless networks: efficient protocols and outage behavior. IEEE Trans. Inf. Theroy. 50(4), 1963–1968 (2004).MathSciNetMATHGoogle Scholar
 B Wang, Z Jun, Z Li, Achieveable rates and scaling laws of powerconstrained wireless sensory relay networks. IEEE Commun.on Infor Theory. 52(9), 4084–4104 (2006).View ArticleGoogle Scholar
 KS Gomadam, SA Jafar, The effect of noise correlation in amplifyandforward relay networks. IEEE Commun.on Infor Theory. 55(2), 731–745 (2009).MathSciNetView ArticleGoogle Scholar
 Y Huang, J Wang, Q Wu, C Zhong, C Li, Outage performance of spectrum sharing systems with MRC diversity under multiple primary user’s interference. IEEE Commun. Lett. 16(4), 576–579 (2014).View ArticleGoogle Scholar
 A Abdi, WC Lau, MS Alouini, M Kaveh, A new simple model for land mobile satellite channels: firstand secondorder statistics. IEEE Trans. Wireless Commun. 2(3), 519–528 (2003).View ArticleGoogle Scholar
 NI Miridakis, DD Vergados, A Michalas, Dualhop communication over a satellite relay and shadowed Rician channels. IEEE Trans. Vehi. Tech. 64(9), 4031–4040 (2015).View ArticleGoogle Scholar
 S Wang, M Yang, Q Guo, in The Fourth Int. Conf. Adv. Sat. Space Commun, Chamonix. Outage probability evaluation of land mobile satellite cooperative diversity communication system (IEEEMontBlanc, 2012), pp. 26–30.Google Scholar
 A Iqbal, KM Ahmed, A hybrid satelliteterrestrial cooperative network over non identically distributed fading channels. J. Commun. 6(7), 581–589 (2011).View ArticleGoogle Scholar
 MR Bhatnagar, MK Arti, Performance analysis of AF based hybrid satelliteterrestrial cooperative network over generalized fading channels. IEEE Commun. Lett. 17(10), 1912–1915 (2013).View ArticleGoogle Scholar
 MR Bhatnagar, MK Arti, Performance analysis of hybrid satelliteterrestrial FSO cooperative system. IEEE Photon. Technol. Lett. 25(22), 2197–2200 (2013).View ArticleGoogle Scholar
 E Costa, S Pupolin, mQAMOFDM system performance in the presence of a nonlinear amplifier and phase noise. IEEE Trans.Commun. 50(3), 462–472 (2002).View ArticleGoogle Scholar
 T Schenk. RF imperfections in highRate wireless systems: impact and digital compensation (SpringerNetherlands, 2008), pp. 2033–2043.Google Scholar
 C Studer, M Wenk, A Burg, in Proc. 2010 ITG Work. Smart Ant. MIMO transmission with residual transmitRF impairments (Bremen, 2010), pp. 189–196.Google Scholar
 J Qi, S Assa, MS Alouini, in Proc. 2012 IEEE Wireless Commun. Netw. Conf. Analysis and compensation of I/Q imbalance in amplifyandforward cooperative systems ((WCNC) PairsFrance, 2012), pp. 1–4.Google Scholar
 E Bjornson, P Zetterberg, M Bengtsson, B Ottersten, Capacity limits and multiplexing gains of MIMO channels with transceiver impairments. IEEE Commun. Lett. 17(1), 91–94 (2013).View ArticleGoogle Scholar
 Z Xinlin, M Matthaiou, E Bjornson, M Coldrey, M Debbah, On the MIMO capacity with residual transceiver hardware impairments (IEEE ICC, Jun 2014 Sydney, Australia, 2014).Google Scholar
 E Bjornson, J Hoydis, M Kountouris, M Debbah, in 2013 8th International Conference on Digital Signal Processing. Hardware impairments in largescale MISO systems: energy efficiency, estimation, and capacity limits (FiraGreece, 2013), pp. 18–20.Google Scholar
 X Xia, D Zhang, K Xu, W Ma, Y Xu, Hardware impairments aware transceiver for fullduplex massive MIMO relaying. IEEE Trans. Sign. Proc. 63(24), 6565–6580 (2015).MathSciNetView ArticleGoogle Scholar
 TT Duy, TQ D, DB Costa, VNQ Bao, M Elkashlan, Proactive relay selection with joint impact of hardware impairment and cochannel interference. IEEE Trans. Commun. 63(5), 1594–1606 (2015).View ArticleGoogle Scholar
 M Maletic, M Cabarkapa, N Neskovic, D Budimir, Hardware impairments impact on fixedgain AF relaying performance in Nakagamim fading. Electronics Lett. 52(2), 121–122 (2016).View ArticleGoogle Scholar
 K An, M Lin, J Ouyang, H Wei, Performance analysis of beamforming in twohop AF relay networks with antenna correlation and interference. AEUInt. J. Electron. Commun. 68(7), 587–594 (2014).View ArticleGoogle Scholar
 C Zhong, S Jin, KK Wong, Dualhop systems with noisy relay and interferencelimited destination. IEEE Trans. Commun. 58(3), 764–768 (2010).View ArticleGoogle Scholar
 YZ Huang, FA Qahtani, C Zhong, Q Wu, J Wang, H Alnuweiri, Performance analysis of multiuser multiple antenna relaying networks with cochannel interference and feedback delay. IEEE Trans. Commun. 62(1), 59–73 (2014).View ArticleGoogle Scholar
 M Sadek, S Aissa, Personal satellite communication: technologies and challenges. IEEE Wireless Commun. 12(5), 28–35 (2012).View ArticleGoogle Scholar
 K An, M Lin, J Ouyang, Y Huang, G Zheng, Symbol error analysis of hybrid satelliteterrestrial cooperative networks with cochannel interference. IEEE Commun. Lett. 18(11), 1947–1950 (2014).View ArticleGoogle Scholar
 8 hints for making and interpreting EVM measurements (2005). Tech. Rep. Agilent Technologies.Google Scholar
 IS Gradshteyn, IM Ryzhik, Table of Intergals Series and Products, 7th ed (Elsevier, Amsterdam, 2005).MATHGoogle Scholar
 The Wolfram Function Site. [Online], Avilable: http://functions.wolfram.com. Accessed 1998.
 A Bletsas, H Shin, MZ Win, Cooperative communication with outageoptimal opportunistic relaying. IEEE Trans. Wire. Commun. 6(9), 3450–3460 (2007).View ArticleGoogle Scholar
 H Shin, MZ Win, MIMO diversity in the presence of double scattering. IEEE Trans. Inform. Theory. 54(7), 2976–2996 (2008).MathSciNetView ArticleMATHGoogle Scholar