There exists a trade-off between energy consumption and spectral efficiency in wireless communication systems under quality of service (QoS) constraints. This paper studies the use of effective capacity theory to characterise the maximum supported channel capacity over fading channels whilst considering both QoS constraints and energy consumption. Moreover, a generalised fading channel model, i.e., the hyper Fox's H fading model, is considered that includes many practical fading channel models as special cases, e.g., Rayleigh, Rician, Weibull and Nakagami-m fading channel models. The results are readily applicable to design energy-aware communication systems over fading channels with QoS constraints, e.g., wireless sensor networks and smart grid communication systems

Sun, Hongjian

You, Minglei

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Minglei You

Hongjian Sun

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Realising energy-aware communication over fading channels under QoS constraints

Durham Research Online

Realising Energy-Aware Communication overFading Channels under QoS ConstraintsMinglei You and Hongjian SunSchool of Engineering and Computing Sciences, Durham University, DH1 3LE, UKEmail: {minglei.you, hongjian.sun}@durham.ac.ukAbstract—There exists a trade-off between energy consumptionand spectral efficiency in wireless communication systems underquality of service (QoS) constraints. This paper studies the use ofeffective capacity theory to characterise the maximum supportedchannel capacity over fading channels whilst considering bothQoS constraints and energy consumption. Moreover, a gener-alised fading channel model, i.e., the hyper Fox’s H fading model,is considered that includes many practical fading channel modelsas special cases, e.g., Rayleigh, Rician, Weibull and Nakagami-𝑚 fading channel models. The results are readily applicableto design energy-aware communication systems over fadingchannels with QoS constraints, e.g., wireless sensor networks andsmart grid communication systems.I. INTRODUCTIONCompared with wired communications, wireless communi-cations have a more flexible and scalable architecture. How-ever, almost all wireless communication networks have tobalance two important aspects, i.e., conserving the limitedsystem resources and providing reliable services. In practice,the energy consumption and available frequency bandwidth arealways confined due to the cost and interference regulation.According to Shannon capacity theory, in order to improvethe throughput under the same channel condition, it has toincrease the transmit power or the frequency bandwidth. Hencethere exists a trade-off between spectral efficiency and energyconsumption. Unfortunately, the traditional Shannon capacitytheory [1] cannot take delay aspects into account to support thecommunication system design in real-time applications, e.g.,smart grid applications (virtual power plant and distributionautomation) that shall consider the time delay introduced bycommunication systems.Wireless communication is not as reliable as its wiredcounterpart, since the wireless channel varies randomly withtime. This fact contributes to the uncertainty in providing QoSguarantee, such as guaranteeing the time delay and providinga minimum throughput. In order to address this issue, thetheory of effective capacity was proposed in [2] that studiesthe maximum supported bit rate over a fading channel understatistical delay constraint. Using the effective capacity theory,the general relation between bit energy consumption andspectral efficiency with QoS constraints was studied [3]. In [1],the general trade-off among energy, bandwidth and delay wasThe research leading to these results has received funding from theEuropean Commission’s Horizon 2020 Framework Programme (H2020/2014-2020) under grant agreement No. 646470, SmarterEMC2 Project.considered, where Rayleigh fading channels were considered.In [4], the problem of maximum the effective capacity withoptimum bit energy over Nakagami-𝑚 fading channels wasanalysed. However, most existing work built on Rayleigh orNakagami-𝑚 fading channels, whereas a general quantifiedanalysis over practical fading channels has been insufficientlystudied.In this paper, we consider the communication systems’ en-ergy performance under statistical QoS constraints over hyperFox’s H fading channels, which is a generalized fading modelincluding many practical fading models as special cases, e.g.,Rayleigh, Rician, Nakagami-𝑚, Weibull, Weibull/Gamma andGeneralized 𝐾 fading model. In this way, the derived quanti-fied analysis of energy performance and spectral efficiencyis readily applicable to the analysis of the aforementionedfading channels, which can be used in designing energy-awarecommunications.II. SYSTEM MODELA. Channel ModelIn this paper, a point-to-point communication system modelsimilar to the one studied in [1] and [4] is considered, which isshown in Fig. 1. We assume that only the receiver has perfectchannel state information (CSI) and the channels are flat blockfading. Thus the channel input-output relation is given by [4]𝑦[𝑗] = ℎ[𝑗]𝑥[𝑗] + 𝑛[𝑗] for 𝑗 = 1, 2, . . . (1)where 𝑥[𝑗] and 𝑦[𝑗] are the channel input and output, and𝑛[𝑗] is the zero-mean, circularly-symmetric, complex Gaussiannoise with variance 𝑁0. Also ℎ[𝑗] denotes the channel fadingcoefficient. The SNR 𝜌 is defined by 𝜌 = 𝑃𝐵𝑁0 , where 𝑃 isthe average transmit power and 𝐵 is the bandwidth. Further,the instantaneous channel gain is denoted by 𝛾[𝑗] = ∣ℎ[𝑗]∣2and the average channel gain is denoted by Ω = 𝔼{∣ℎ[𝑗]∣2}.For notational brevity, the time index 𝑗 is omitted.B. Hyper Fox’s H fading channelHyper Fox’s H fading model is a generalized fadingmodel [5], which can exact represent or approximate variouspractical fading models as special cases, such as Rayleigh,Rician, Weibull, Nakagami-𝑚, Lognormal, Generalized 𝐾,Weibull/Gamma and standard Fox’s H fading model. Let 𝛾be a random variable following the hyper Fox’s H fadingData Source QoS Constraint  Transmitter Receiver  with CSI Fading Channel y[j]=h[j]x[j]+n[j] Data Sink x[j] y[j] Fig. 1. System modeldistribution, then its PDF is given by [5]𝑓(𝛾) =𝐾∑𝑘=1𝐻[𝛾,O𝑘,P𝑘], (2)where {O𝑘=(𝑚𝑘, 𝑛𝑘, 𝑝𝑘, 𝑞𝑘),P𝑘=(𝑢𝑘, 𝑣𝑘, c𝑘,d𝑘,C𝑘,D𝑘)(3)whereas c = (𝑐1, . . . , 𝑐𝑝),d = (𝑑1, . . . , 𝑑𝑞),C =(𝐶1, . . . , 𝐶𝑝) and D = (𝐷1, . . . , 𝐷𝑞). The parameters 𝐾,O𝑘 and P𝑘 satisfy the conditions given in [5]. The Fox’s Hfunction is defined by a single Mellin-Barnes type of contourintegral as [6, Ch. 1.2]𝐻 [𝛾,O,P] =𝑢2𝜋𝑖∮ℒΦ(𝑧) (𝑣𝛾)−𝑧𝑑𝑧, (4)where 𝑖 =√−1, 𝑧 ∕= 0 and ℒ is a suitable contour. Thefunction Φ(𝑧) is defined by [6]Φ(𝑧)=∏𝑚𝑗=1 Γ (𝑑𝑗+𝐷𝑗𝑧)∏𝑛𝑗=1 Γ (1−𝑐𝑗−𝐶𝑗𝑧)∏𝑝𝑗=𝑛+1 Γ (𝑐𝑗+𝐶𝑗𝑧)∏𝑞𝑗=𝑚+1 Γ (1−𝑑𝑗−𝐷𝑗𝑧). (5)Further, we define the following functionΨ(𝑧) = 𝑢𝑣−𝑧Φ(𝑧) , (6)which will be used to deduce the spectral efficiency metric inthe following part.C. Effective capacity theoryEffective rate is the maximum constant rate that a fadingchannel can support under statistical delay constraints. Theeffective capacity of the service process can be written as [2]𝛼(𝜌, 𝜃) = − 1𝜃𝑇ln𝔼{𝑒−𝜃𝑇𝐶}, 𝜃 ∕= 0, (7)where 𝐶 represents the system’s throughput during a singletime block and 𝑇 denotes the duration of a time block. TheQoS exponent 𝜃 is given by [2]𝜃 = − lim𝑧→∞lnPr{𝐿 > 𝑧}𝑧, (8)where 𝐿 is the equilibrium queue-length of the buffer at thetransmitter, and Pr{⋅} is the probability of an event. The QoSexponent 𝜃 has to satisfy the constraint 𝜃 ≥ 𝜃0, where 𝜃0 isthe minimum required QoS exponent. Larger 𝜃0 correspondsto a tighter delay constraint, whereas 𝜃0 = 0 implies nodelay constraints. Also when 𝜃0 → 0, the effective capacityapproaches the Shannon’s channel capacity [7].III. EFFECTIVE CAPACITY OVER GENERAL FADINGCHANNELSAccording to Shannon’s theory, if we want to improve thespectral efficiency, i.e. transmit more bits under the samebandwidth, a direct way is to increase the SNR, which canbe achieved by increase the transmit power. But in practical,increase the transmit power will usually result in higher energyconsumption and more interference to other transceivers. Inthis section, we will quantify the relation between bit energyand spectral efficiency under a dedicated QoS constraint.In this paper, the energy per information bit 𝐸𝑏/𝑁0 (indB) is used as the metric for energy performance, which canprovide a unified way to measure and compare the energy per-formance between different wireless communications systems[4]. Moreover, effective capacity normalized by bandwidth𝑅(𝜌, 𝜃) (in bit per second per hertz) is used as the metric forspectral efficiency. Specifically, the spectral efficiency metricis defined as [1]𝑅(𝜌, 𝜃) =𝛼(𝜌, 𝜃)𝐵= − 1𝜃𝑇𝐵ln𝔼{𝑒−𝜃𝑇𝐶}. (9)The effective capacity is a concave function of the SNR 𝜌and the minimum bit energy is achieved as 𝜌 → 0 [1], wherethe effective capacity can be approximated by𝑅(𝐸𝑏𝑁0, 𝜃)≈ 𝑆0 log2(𝐸𝑏𝑁0𝐸𝑏𝑁0 min), (10)where 𝑆0 denotes the capacity slope in bits/s/Hz/(3dB) and𝐸𝑏/𝑁0min is the minimum energy per information bit requiredunder QoS constraints, which are given by [1]⎧⎨⎩𝐸𝑏𝑁0 min= lim𝜌→0𝜌𝑅 (𝜌, 𝜃)=1𝑅′ (0, 𝜃)𝑆0 = −2 [𝑅′ (0, 𝜃)]2 ln 2𝑅′′ (0, 𝜃)(11)where 𝑅′ (0, 𝜃) and 𝑅′′ (0, 𝜃) denote the first and second orderpartial derivatives with respect to 𝜌 evaluated at 𝜌=0.When considering hyper Fox’s H fading channels, thesemetrics can be calculated via the following theorem.Theorem 1: When only the receiver has perfect CSI, the bitenergy, spectral efficiency and QoS exponent can be quantifiedby (10), with the minimum bit energy 𝐸𝑏𝑁0 min and capacity slope𝑆0 under hyper Fox’s H fading channel given by⎧⎨⎩𝐸𝑏𝑁0 min=ln2Ω𝑆0 =2(𝐴+1)Ω2∑𝐾𝑘=1 Ψ𝑘(3)−𝐴(12)where 𝐴 = 𝜃𝑇𝐵ln2 and the function Ψ(𝑧) is defined in (6).Proof: By calculating the first and second order partial deriva-tives with respect to 𝜌 using (9), we have 𝐸𝑏𝑁0 min =ln2𝔼{𝛾} and𝑆0 =2(𝐴+1)𝔼{𝛾2}/(𝔼{𝛾})2−𝐴 [1, eq.(18)]. Then using Mellintransform [6, eq.(2.8)] and (6), (12) can be achieved.Remark: It can be seen that the minimum required bitenergy is only related to the average channel gain and it doesnot depend on specific channel fading parameters, which isconsistent with [1]. The capacity slope 𝑆0 depends not onlyon the QoS requirement 𝜃, but also fading parameters. Inthis context, the bit energy can be adjusted according to thecommunication requirements that realises the energy-awarecommunication while providing statistical QoS guarantees.As discussed in Section II-C, when 𝜃 = 0, there is no delayrequirement on the transmission. Hence the effective capacitybecomes Shannon capacity, and the metrics can be used intraditional Shannon capacity analysis by letting 𝜃 = 0, whichcan be calculated by the following corollary.Corollary 1: When only the receiver has perfect CSI andthere is no delay constraint, the minimum bit energy andcapacity slope under hyper Fox’s H fading channel reduce to⎧⎨⎩𝐸𝑏𝑁0 min=ln2Ω𝑆0 =2Ω2∑𝐾𝑘=1 Ψ𝑘(3)(13)Proof: These results can be obtained by substituting 𝜃 = 0into Theorem 1.Hyper Fox’s H fading model is a general model, but itsparameters have physical meaning only when specific fadingmodel is considered. Hence in the following, we considerseveral typical and practical fading models, namely Rayleigh,Weibull, Nakagami-𝑚, Weibull/Gamma and Generalized 𝐾fading channels as examples to show the usage of Theorem1. All these fading models are special cases of hyper Fox’s Hfading model.∙ Rayleigh fading: If there are only non-light-of-sight paths,measurements show that the Rayleigh distribution cancharacterize the statistical distribution of the receivedsignal’s envelop [8]. The instantaneous SNR parametersequences for Rayleigh distribution are [8, Table IX]O = (1, 0, 0, 1) and P = (1, 1,—, 12 ,—,12 ). By usingTheorem 1, the capacity slope can be given by 𝑆0 = 2𝐴+2 .∙ Nakagami-𝑚 fading: The Nakagami-𝑚 distribution hasbeen first introduced in [9], which can characterize theenvelop of the received signal propagating in ionosphericand tropospheric environments. The instantaneous SNRparameter sequences are [8, Table IX] O = (1, 0, 0, 1)and P = ( 𝑚Γ(𝑚) ,𝑚,—,𝑚− 1,—, 1), where 𝑚 describesthe fading severity. The capacity slope can be given by𝑆0 =2𝑚𝐴+𝑚+1 . The Nakagami-𝑚 fading includes theRayleigh fading (𝑚 = 1) as special cases and it canapproximate Nakagami-𝑞 (or Hoyt) fading (𝑚 < 1) andRician fading (𝑚 > 1).∙ Weibull fading: Weibull fading model can characterize theeffect of clusters of multipath waves propagating in non-homogeneous environment [7]. The instantaneous SNRparameter sequences are [8, Table IX] O = (1, 0, 0, 1)and P = (Γ(1 + 2/𝛽),Γ(1 + 2/𝛽),—, 1 − 2/𝛽,—, 1 −2/𝛽), where 𝛽 describes fading severity. The capacityslope can be given by 𝑆0 = 2Γ(4/𝛽+1)(𝐴+1)Γ2(2/𝛽+1)−𝐴 . This modelincludes Rayleigh fading (𝛽 = 2) and exponential fading(𝛽 = 1) as special cases.∙ Weibull/Gamma fading: Weibull/Gamma fading modelcharacterizes the composite effect of Weibull multipathfading and gamma shadowing [10]. The instantaneousSNR parameter sequences for Weibull/Gamma fadingmodel are [8, Table IX] O = (2, 0, 0, 2) and P =(Γ(1+2/𝛽)𝜓Γ(1/𝜓) ,Γ(1+2/𝛽)𝜓 , –, (1𝜓 − 1, 1 − 2𝛽 ), –, (1, 2𝛽 )), wherethe fading severity parameter 𝛽 > 0 and shadowingfigure 𝜓 ≥ 0. The capacity slope can be given by𝑆0 =2𝜓Γ(4/𝛽+1)(1/𝜓+1)(𝐴+1)Γ2(2/𝛽+1)−𝐴 . This model reduces toWeibull fading model when 𝜓 = 0.∙ Generalized 𝐾-fading: Generalized 𝐾-fading model hasbeen first introduced in [11] to model the intensity ofradiatio scattered with a non-uniform phase distribution,which accounts for the composite effect of Nakagami-𝑚 multipath fading and gamma shadowing. The instan-taneous SNR parameter sequences for generalized 𝐾-fading model are [8, Table IX] O = (2, 0, 0, 2) andP = ( 𝑚𝜓Γ(𝑚)Γ(1/𝜓) ,𝑚𝜓 , –, (𝑚−1, 1𝜓 −1), –, (1, 1)), wheremultipath fading severity parameter 𝑚 ≥ 12 and shadow-ing figure 𝜓 ∈ [0, 2]. The capacity slope can be given by𝑆0 =2(𝑚+1)(1/𝜓+1)𝑚/𝜓(𝐴+1)−𝐴 . Specially, when 𝜓 = 0, thegeneralized 𝐾-fading reduces to the Nakagami-𝑚 fading.Here we list the capacity slope 𝑆0 of the aforementionedtypcial fading channels in Table I. It can be verified that when𝜃 = 0, these results coincide with [8].TABLE ICAPACITY SLOPE 𝑆0 FOR TYPICAL FADING CHANNELSFading Channel 𝑆0Rayleigh 2𝐴+2Nakagami-𝑚 2𝑚𝐴+𝑚+1Weibull 2Γ(4/𝛽+1)(𝐴+1)Γ2(2/𝛽+1)−𝐴Weibull/Gamma 2𝜓Γ(4/𝛽+1)(1/𝜓+1)(𝐴+1)Γ2(2/𝛽+1)−𝐴Generalized 𝐾 2(𝑚+1)(1/𝜓+1)𝑚/𝜓(𝐴+1)−𝐴IV. NUMERICAL RESULTSIn order to verify the the quantified relation among energyperformance, spectral efficiency and QoS exponent, simula-tions under Weibull/Gamma fading channels have been carriedout. The spectral efficiency is approximated using (10), wherethe parameters are calculated using Table I. Without loss ofgenerality, the duration of a time block 𝑇 = 1 ms and thebandwidth of the system 𝐵 = 1 kHz are assumed. For eachsimulated scenario, we run 107 trails with unit power.−2 −1.5 −1 −0.5 000.20.40.60.8Eb/N0(dB)Spectral Efficiency(bit/s/Hz)  ApproximationSimulationβ=3, ψ=0, θ=1β=3, ψ=0, θ=0β=3, ψ=0.5, θ=1Fig. 2. Spectral efficiency versus energy performance over Weibull/Gammafading channels.It can be seen that the proposed approximation fits tightwith the simulation resutls in Fig. 2. The results in the case𝛽 = 3, 𝜓 = 0, 𝜃 = 0 denote the Shannon capacity underWeibull fading channels, which agree with the results in [8]and support the validation of our derivation. With the sameFig. 3. Required bit energy for different fading severity 𝛽 and shad-owing figure 𝜓 in Weibull/Gamma fading channels (Spectral efficiency0.1 bit/s/Hz,𝜃=1)spectral efficiency and QoS constraint, the bit energy underdifferent fading severity and shadowing figure is shown inFig. 3. The increase of multipath fading is indicated by thedecrease of fading severity 𝛽, while the shadowing gets severindicated by the increase of the shadowing figure 𝜓. It canbe seen from Fig. 3, in order to fight against the fading effectcaused by multipath fading and shadowing, the energy requiredto maintain the same spectral efficiency is increased in anexponential way.It gives a more general view of the effect of QoS require-ment 𝜃 and energy efficient metric 𝐸𝑏/𝑁0 on the spectralefficiency in Fig. 4. It is indicated from both Fig. 2 and Fig. 4that under the same bit energy, the spectral efficiency decreasesas the delay requirement becomes more strigent, which iscorresponding to the increase of QoS requirement 𝜃. Whenthe fading parameters are known as a prior, the quantifiedrelation as proposed in Theorem 1 can be used in the study oftrade-offs between spectral efficiency and energy performance.Fig. 4. Spectral efficiency v.s. bit energy 𝐸𝑏/𝑁0 and QoS exponent 𝜃 inWeibull/Gamma fading channels (𝛽=3,𝜓=0.5)V. CONCLUSIONIn this paper, the relationship between bit energy andspectral efficiency has been studied considering the hyper Hfading channel and QoS requirements. The results have shownthat the proposed approach is readily applicable to variousfading channels. The bit energy consumption decreases as theQoS requirement increases. It is noteworthy highlighting thatby using the quantified relationship among bit energy, spectralefficiency and QoS requirement, we can adjust the energyconsumption of a wireless communication system accordingto the QoS requirement and channel condition. This result canbe used as an important foundation to realise energy-awarecommunications in real-time applications over fading channelswith QoS constraints.REFERENCES[1] M. C. Gursoy, D. Qiao, and S. Velipasalar, “Analysis of energy efficiencyin fading channels under QoS constraints,” IEEE Trans. Wirel. 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Realising energy-aware communication over fading channels under QoS constraints

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