Cross-Layer Optimization of Wireless Links under Reliability and Energy Constraints

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1 Cross-Layer Optiization of Wireless Links uner Reliability an Energy Constraints Aair Mahoo, M M Aftab Hossain an Mikael Gilun Departent of Inforation Systes an Technology, Mi Sween University, Sween Wireless@KTH, KTH Royal Institute of Technology, Sween Eail: firstnae.lastnae@iun.se, aho@kth.se arxiv:75.774v [cs.it] 5 Sep 7 Abstract The vision of connecting billions of battery operate evices to be use for iverse eerging applications calls for a wireless counication syste that can support stringent reliability an latency requireents. Both reliability an energy efficiency are critical for any of these applications that involve counication with short packets which unerine the coing gain achievable fro large packets. In this paper, we stuy a cross-layer approach to optiize the perforance of lowpower wireless links. At first, we erive a siple an accurate packet error rate PER expression for uncoe schees in block faing channels base on a new proposition that shows that the waterfall threshol in the PER upper boun in Nakagai- faing channels is tightly approxiate by the -th oent of an asyptotic istribution of PER in AWGN channel. The propose PER approxiation establishes an explicit connection between the physical an link layers paraeters, an the packet error rate. We exploit this connection for cross-layer esign an optiization of counication links. To this en, we propose a sei-analytic fraework to jointly optiize signal-to-noise ratio SNR an oulation orer at physical layer, an the packet length an nuber of retransissions at link layer with respect to istance uner the prescribe elay an reliability constraints. I. INTRODUCTION The upcoing wireless networks are require to support assive nuber of evices uner the ubrella of internet-ofthings IoT. The heterogeneity of use cases of these achineto-achine MM type counication necessitate iverse reliability an latency requireents. As the evices will be ostly battery operate, energy efficiency also becoes a critical issue. Moreover, this novel traffic type uses short packets which unerine the coing gain achievable fro large packets [] []. All these factors urge a new look not only into physical layer but also cross-layer esign in orer to ensure reliability uner energy constraints. In this paper, we iprove the packet error rate PER approxiations over block-faing channels in orer to have a better control over the paraeters that eterine the syste perforance an utilize these insights to optiize cross-layer paraeters, e.g., packet length, nuber of retransission, oulation schee. The average PER is an iportant etric for cross-layer optiization of wireless transissions over block-faing channels. For instance, the objective function to optiize throughput, energy or spectral efficiency of a transission schee is efine in relation to the average PER [3] [4], an the paraeters axiizing the syste perforance are eterine. However, the average PER, except for certain siple cases, is not foun in exact close for, although it can usually be written in the integral for. The integral then nees to be evaluate nuerically an ay not be coputationally intensive, however this approach in general oes not offer insights as to what paraeters eterine the syste perforance. One such close for is the upper boun on average PER for both the uncoe an coe schees in Rayleigh faing, expω /, where is the average signal-to-noise ratio SNR an ω is the waterfall threshol [5]. The threshol is efine as an integral of the PER function in the AWGN channel. In [5], also a siilar upper boun in Nakagai- faing is propose with a corresponing threshol. In both cases, however, a close-for solution to the threshol is not feasible. A log-oain linear approxiation of ω is evelope for uncoe schees in [6], an for uncoe schees in [7]. However, the approxiation in [6] is tight for large packets only while in [7] the approxiatioaraeters for a given oulation schee are calculate by siulations. For uncoe schees, an accurate PER expression is erive in [8] that is coplicate to utilize for link optiization. In this paper, we show that the waterfall threshol in the PER upper boun uner Nakagai- block faing is tightly approxiate by the -th oent of an asyptotic istribution of PER function in AWGN. In Rayleigh faing, the approxiation leas to a PER approxiation which is accurate than [6] [7] an also aintains explicit connection with oulation orer unlike [7]. Note that in toay s age of battery operate evices, the nuber one esign goal is energy efficient counication. However, the eerging elay an reliability requireents have a irect ipact on the neee energy to transfer each inforation bit. The reliability epens on the bit error or packet error statistics of the wireless channel, which in turn epen on the choice of the syste paraeters such as transit power, oulation schee, packet length etc. If the packet error probability has to be reuce so as to transfer a packet with liite nuber of retransission, these paraeters nee to be optiize while keeping a tab on the energy consuption. How to select the oulation orer an transissioower to attain energy-efficient counication is stuie in AWGN channel [9] [] an in faing channels [4] [7]. These stuies in general suggest using higher orer oulations at saller istances as oppose to the coon notion followe in wireless sensor networks WSN by choosing low-orer

2 oulations for their low SNR requireent. For instance, lowpower transceivers CC an CC4, often use in WSNs, eploy BPSK an QPSK. In faing channels, it is shown in [4] [7] that there exist an optial SNR an packet length for each oulation schee at which the require energy for successful transfer of an inforation bit is iniize. In [4], the optial SNR is conitione on the axiu transit power however this constraint is ignore in [7]. In these stuies, no restriction on nuber of retransission is ipose an as a result the optial SNR is not boun to satisfy the reliability target. In this paper, we stuy the energy iniization in faing channels however uner the often neglecte reliability constraints. We exploit the propose PER approxiation for cross-layer optiization of a power-liite syste in Rayleigh block-faing channels. By efining a energy consuption oel for per payloa bit transferre, we fin the optial energy consuption iniizing syste paraeters while aintaining the reliability an elay target. Specifically, i for a syste with fixe oulation schee e.g., CC4 an report size, we propose close-for conitions for energy optial SNR that confor to the axiu transit power an reliability constraints, ii for a general power-liite syste, we propose a joint optiization algorith to fin the physical layer SNR, oulation orer an link layer packet length, nuber of retransissions paraeters with respect to istance uner the prescribe elay an reliability constraints. The rest of the paper is organize as follows. Section II evelops an approxiation to average PER in block faing channels. Section III efines the cross-layer optiizatioroble, solves it uner the reliability constraints an presents the results. Section IV raws the concluing rearks. II. THE AVERAGE PER IN BLOCK FADING Let f γ be the PER function in the AWGN channel with instantaneous SNR, γ. Then, for ann-bit uncoe packet with bit error rate BER function b e γ, f γ is efine as N fγ = b e γ Also, let pγ; be the probability istribution function PDF of the receive instantaneous SNR. In Nakagai- faing, γ follows the Gaa istribution with PDF pγ; = γ Γ exp The average PER, enote as integrating over P e γ = γ,γ P e, is then copute by fγpγ; γ 3 In [5], it is shown that Pe is upper boune by P e γ B exp ω Γ B where γ f γ B an ω is efine as, ω = 4 γ f γγ 5 In Rayleigh faing i.e., =, as f γ is the probability we have f γ, an 4 can be written as, P e γ exp ω 6 where ω fro 5 becoes ω = f γγ 7 In what follows, we propose generic approxiations to ω an ω for uncoe schees with BER functions as b e γ = c exp k γ 8 b e γ = c Q k γ 9 where c an k are oulation-epenent constants. Noncoherent FSK an DPSK have the BER in the for of 8 while M-ASK, M-PAM, MSK, M-PSK an M-QAM have BER in the Gaussian Q-function for 9 [5]. A. Approxiations to ω an ω Proposition : For uncoe transission of a packet with length N, with the BER functions escribe by c e kγ an c Q k γ where < c an k >, the threshol, ω, in Nakagai- faing channel for integer values of the faing paraeter is approxiate by the th oent of the Gubel istribution for saple axiu ω E[γ ] Proof : For packet length N, the PER function in for BER functions escribe by c e kγ an c Q k γ can be asyptotically approxiate by the Gubel istribution function for the saple iniu [8] fγ exp exp γ a N b N where a N an b N > are the noralizing constants. Let Gγ = exp exp γ an b N be the cuulative istribution function CDF of the Gubel istribution for the saple axiu, then fro an 5 we have ω γ Gγ γ Assuing γ = y an γ γ = y/, fro we get ω G y y 3 Let gγ = Gγ/γ be the PDF, then with soe anipulation an changing the orer of integration in 3 ω = = y γ gγγy gγyγ γ gγγ. 4

3 Noting that the integral in the last equality is the th oent of a continuous an nonnegative rano variable γ with the PDF gγ copletes the proof. One can fin the th oent of the Gubel istribution fro its oent generating function MGF efine as M γ t Γ b N t e an t 5 where Γ is the stanar gaa function. In Rayleigh faing with =, fro an 5, ω equals the expecte value of the Gubel istribution, i.e. ω E[γ] = a N +γ e b N 6 where γ e =.577 is the Euler constant. Notation ω is preferre over ω to reain consistent with the prior works. Siilarly, for = an = 3, which represent the next two significant faing conitions, uner 5 becoes ω E[γ ] = ] [a N +.64b N +γ eb N +γ e a N b N 7 ω 3 E[γ3 ] = [ 4.93γ e b 3 N a Nb N +a3 N +.4b3 N ] 8 +γeb 3 3 N +3γea N b N +3γ e a Nb N The noralizing constants for BER function in 8 are [8] a N = lognc k, b N = k 9 whereas the constants for BER in 9 are a N = [ erf ] k Nc b N = k [ erf Nc e where e is the base of the natural logarith. B. The Average PER with New Paraetrization ] an Using the noralizing constants 9 in, the average PER in 4 an 6 can be expresse in the for of eleentary functions. However ue to the inverse error function in, 4 an 6 cannot be siplifie further. An intuitive approach is to utilize an exponential function base approxiation of Q-function e.g., [], an utilizea N anb N fro 9. However, this approach loses the approxiation accuracy. Instea, our objective is to fin the exponential function approxiation for given a N an b N in 9 an ω approxiation in that fits best to the integral expression in 5 or 7. In essence, we reforulate a N an b N in 9 as a N logk Nc k k, b N k k an fin the constants k an k. We estiate k an k for BPSK oulation by nuerically evaluating 7 an atching it with 6 uner a N an b N in. For a packet length N in an interval [3,4] bits, the optial constants are: k =.4 an k = We fin that these constants are inepenent of oulation schees with the BER function involving Q-function. As a result, a siple PER approxiation Relative Error % Relative Error % N = N = Upper boun 6 Propose approx..5 Propose approx. Wu et al. [7] approx. Liu et al. [6] approx a 4 QAM N = Upper boun 6 Propose.5approx. Propose approx. Wu et al. [7] approx. Liu et al. [6] approx b 6 QAM N = Fig.. Relative error in average PER in Rayleigh faing. Approx. uses a N an b N fro while Approx. is base on. can also be reache for the oulation schees with the BER functionc Q k γ. For instance, let ć = k c anḱ = k k, then fro an 6, the PER in Rayleigh faing is P e Nć ḱ exp γ e ḱ. whereć = c anḱ = k for the BER functionc e kγ. We evaluate the average PER in 4 with the propose ω approxiation in with the original an oifie paraeters an valiate it against the nuerical evaluation of PER in 3. The approxiations to 4 for Rayliegh faing channel propose in [6] [7] are also analyze. Fig. a an Fig. b show the relative error RE in the propose approxiation an the reference stuies for 4-QAM an 6- QAM in Rayleigh faing. Siilar results not shown here are obtaine for 64-QAM. It can be observe that the RE in average PER with ω approxiations an are quite close to the upper boun 6, which is evaluate nuerically, for sall to large packet lengths. In coprison, the RE of approxiations in [6] [7] is sall at low SNR, however it increases rapily especially for sall packet lengths. We also evaluate the average PER base on ω approxiations in 7 an 8 an observe the siilar accuracy.

4 III. ENERGY EFFICIENT LINK OPTIMIZATION A. Energy Consuption Moel We consier iniizing energy consuption of a wireless link between a transitter an receiver pair separate by istance. The energy consuption of the signal path at the transitter an receiver is coprise of baseban processing blocks e.g., ecoing an eoulation an raio-frequency RF chain that consists of a power aplifier PA an other electronic coponents such as analog-to-igital an igital-to-analog AD/DA converter, low-noise aplifier LNA, filters, ixers an frequency synthesizers. However for an energy-constraine wireless syste e.g., WSN, the energy consuption of RF chain is orers of agnitue larger than that of baseban processing coponents. The power consuption of PA is consiere to be proportional of the transit power P t such that P PA = ξ η P t, where η is the rain efficiency of the power aplifier PA an ξ is the peak-to-average-power-ratio PAPR. The PAPR epens on the oulation schee an the associate constellation size. If baseban power consuption is neglecte an the power consuption of all the other coponents in RF chain excluing PA is enote as P c, a siple power consuption oel is P on = ξ η P t +P c. Fro [9], this oel leas to the total energy consuption to transit an receive a sybol as E sy = ξ η E t + P c R s 3 where E t is the average transission energy of a sybol an R s the physical layer sybol rate. For FSK, BPSK an QPSK oulations ξ =, for OQPSK ξ =.38, an for a square MQAM oulation ξ = 3 M M + [9]. Let E b = E r /log M be the average receive energy per uncoe bit where E r is the average receive energy per sybol an M is the constellation size, then the average SNR,, at the receiver is = E r N log M 4 Assuing a κth-power path-loss oel, the transission energy at istance fro 4 is expresse as [9] E t E r G = N log M G 5 where G G κ M l is the pathloss gain with G, the gain factor at unit istance, epens on the transit an receive antenna gains an carrier frequency, an M l the link argin. Iacket base wireless systes, the inforation bits are encapsulate into packets each carrying payloa an n h overhea bits. The nuber of sybols in a packet are n s = n h + /log M. The average energy require to transit an receive an inforation bit per packet transission, fro 3 an 5, is E = n s E sy = +n h A +B 6 where A = ξn G /η an B = ns Pc R s = Pc R b with R b = W log M the physical layer bit rate in banwith W. The total energy consuption of a wireless link epens on the require retransissions before a packet is ecoe successfully at the receiver. The retransission statistics are eterine by the PER, P e, which is a function of, channel faing, an other paraeters as iscusse earlier. The nuber of retransissions τ is geoetric rano variable an over an uncorrelate channel between retransissions the average nuber of retransissions are τ = / P e. Therefore, the total average energy for a successful transission of a bit is E = τe, which fro 6 is E = P e np +n h A +B 7 In forulating 7, no liit on the nuber of retransissions is assue. However for a elay constraine syste, a packet ust be elivere within axiu nuber of retransissions τ ax r τ ax r an the packet error probability after retransissions ust be less than a reliability target ε [ Pe ] τ ax r + ε 8 Fro 8, the require PER ε req to satisfy target ε is P e ε /τax r + := ε req 9 If 9 is satisfie, the average nuber of transissions per packet is τ trunc = [ P e ] τax r + / P e an the total average energy is given by E trunc = [ P e ] τax r + np +n h P A +B 3 e In next section, we consier iniizing energy consuption per inforation bit in 7 while aintaining the PER constraint in 9. B. Link Optiization with Miniu Energy Consuption Optial Average SNR: With fixe, fining the optial average SNR represents a case where the sensors have to sen a fixe size reports. The unconstraine energy iniization proble for optial is oele as iniize subject to E [, ] 3 The function E is a prouct of two functions: τγ the nuber of retransissions with τ γ, an E γ the average energy per transission attept such that E γ where x enotes the first erivative. If both τγ ane γ are convex, then E is also convex [7, Lea ] an the optial can be obtaine by solving E = which yiels a quaratic equation with a positive root as = ω + ω ω 4 + B A n h + 3 Uner the constraints on require PER an the transit power, the iniization of energy in 7 can be written as iniize E 33 subject to in ax

5 The iniu average SNR in requireent is set by the PER boun in 9, which can be obtaine fro γ e +log ć n h + in = 34 ḱ log ε req Due to the harware an regulatory constraints, the transission power cannot excee a liit P. The conition P tx P translates to ax with ax fro 5 is ax = P WN G log M 35 Fro 34 an 35, the require SNR, enote as req, relates to the SNR for unconstraine case in 3 as in, < in req = ax, > ax 36, otherwise which hols for in < ax. If in > ax, the reliability target cannot not be satisfie for a given oulation schee. Optial Payloa Size: The function E in 7 is also convex iayloa size an its optial value is n h ḱ + ḱ +ḱ+ 4Bk n A + p = 37 + B A The upper liit on the payloa size,ax is set by the iniu SNR requireent in to satisify PER target. It is given by fro 9,ax = n h + γe+inḱ log εreq ć 38 where in is given in Joint Optial,,M,τr ax : As the IoT evices will be use in iverse scenarios, it ight be iportant in any to fin the optial SNR, payloa size, oulation orer an nuber of retransissions for energy efficient counication. For exaple after eployent in harsh an inaccessible areas, the evices will optiize those paraeters for the first tie an then can continue with the optial setting. The joint optiizatioroble can be written as iniize,,m,τ ax r E,,M,τ ax r 39 where M {FSK,MPSK,MQAM} an τr ax = i, i. Note that these evices will support only few values of M an a sall value of τr ax is feasible for iniu energy operation [7]. As a result, the exhaustive search over the cobination of M an τr ax will not be coputationally eaning. For each cobination of M an τr ax, the joint optiu an can be foun fro 3 an 37 either by solving syste of two non-linear equations or by iteratively invoking these equations. In either case, we nee to ensure that the reliability conitions in 36 an 38 are satisfie. However, the forer etho requires nuerical evaluation that ight be coputationally infeasible for harware-constraine Algorith Joint Optiization with Reliability Target Input: ε req,τr ax,δ Output:,n p,τr,m : for M [FSK,MPSK,MQAM] o : for i = to τr ax o 3: 4: while > δ o 5: Evaluate 3, in Evaluate 34, ax Evaluate 35,,ax Evaluate 38 6: if in > ax then 7: break; 8: else 9: req Evaluate 36 : en if : Evaluate 37 with = req : if >,ax then 3:,ax 4: en if 5: E Evaluate 3 Print E,γ,,τ r,m 6: req, = req 7: en while 8: en for 9: en for : return,,τ r,m yieling iniu E evices. On the other han by iteratively invoking 3 an 37, an can efficiently converge to joint energy optiu values while satisfying the reliability conitions. It is straightforwar to evelop the proof of convergence of the iterative approach by following [7, Corollary 3]. Note that by initializing an to any value, this approach converges within a few iterations to optiu values. A pseuocoe of the propose joint optiization is given in Algorith. C. Nuerical Results The siulatioaraeters are taken fro [9]: N / = 74 B/Hz, κ = 3.5, G = 3B, M l = 4B, W = khz, P c {MQAM,MPSK} = 3W, Pc FSK = 65W, η = 35%. Other paraeters are: = 48 bits, ε =. i.e., 99.9% reliability, P = W. Fig. shows an exaple case of iniu SNR in require at various istances. The unconstraine optial SNR an axiu achievable SNR ax are also epicte. At =, in is less than, therefore is energy optial an is preferre over in. While at = 3, cannot satisfy the target an in, though not energy optial, is selecte. At = 7, the reliability target is not satisfie as in > ax. In Fig. 3, energy consuption for selecte oulation schees with respect to istance when operate at optial require SNR req is shown. The conition at which in cannot be satisfie at a given transit power constraint is also epicte. In aition, for in > ax, we set γ req = γ ax to epict the energy consuption uner unliite retransissions. It is observe that there is an optial oulation schee at each istance that also satisfies the reliability target: high-orer oulations at lower istance an low-orer at higher istance as shown without reliability constraints in [4]. However for given transit power liit, the istance at which the reliability target is satisfie ecreases as the reliability requireent becoes tight.

6 Energy consuption/bit BJ a c 3 64 QAM 6 QAM OQPSK BPSK 5 b Fig.. Optial SNR vs require SNR for 4-QAM uner reliability constraints of ε =., τr ax = 3 an axiu transit power of P = W. Energy consuption/bit BJ Fig. 3. Energy consuption of the oulation schees with optial require SNR at each istance. The arke conition in ax shows the istance beyon which the reliability constraints are not satisfie. Siulation paraeters: P = W, ε =., τr ax = 3, = 984, n h = 4. In Fig. 4, one can grasp the big picture of how the paraeters,,m an τr ax vary with istance. At very short istance high M an lower are energy efficient. The reason behin lower can be explaine with the saller value of τr ax. As istance increases optial M becoes saller. The payloa size keeps increasing at aroun7 8 an 3 9 region with the increase in istance keeping the alost constant, i.e., increasing transit power with increasing packet size is optial until next saller M becoes energy optial. At long istances lowerm an are energy optial. IV. CONCLUSIONS In this paper, we stuie the cross-layer link optiization while ensuring energy efficiency an reliability constraints. For cross-layer analysis, we first presente a siple approxiation to average PER in block faing channels. The propose PER approxiation is in the for eleentary functions, an aintains an explicit connection between the physical/link Fig. 4. Optial paraeters at the output of the joint optiization algorith: a energy consuption, b payloa size bits, c SNR, nuber of retransissions. The siulatioaraeters are the sae as in Fig. 3. layer paraeters an the packet error rate. The nuerical analysis confirs the tightness of the approxiation as copare to earlier stuies. Later, we exploite the propose PER approxiation in the energy consuption oel to fin energy optial yet reliability an harware copliant conitions for unconstraine optial SNR an payloa size. These conitions are shown to be useful to: i fin optial SNR for a syste with fixe oulation schee an payloa size, ii evelop an holistic algorith to jointly optiize the physical an link layer paraeters. REFERENCES [] G. Durisi, T. Koch, an P. Popovski, Towar assive, ultrareliable, an low-latency wireless counication with short packets, Proc. of IEEE, vol. 4, no. 9, pp. 7 76, Sept 6. [] C. E. Shannon, A atheatical theory of counication, Bell Syst. Tech. J., vol. 7, no. 3, pp , 948. [3] Q. Liu, S. Zhou, an G. B. Giannakis, Cross-layer cobining of aaptive oulation an coing with truncate ARQ over wireless links, IEEE Trans. W. Coun., vol. 3, no. 5, pp , 4. [4] F. Rosas an C. Oberli, Moulation an SNR optiization for achieving energy-efficient counications over short-range faing channels, IEEE Trans. W. Coun., vol., no., pp ,. [5] Y. Xi, A. Burr, J. Wei, an D. Grace, A general upper boun to evaluate packet error rate over quasi-static faing channels, IEEE Trans. W. Coun., vol., no. 5, pp ,. [6] S. Liu, X. Wu, Y. Xi, an J. Wei, On the throughput an optial packet length of an uncoe ARQ syste over slow Rayleigh faing channels, IEEE Coun. Lett., vol. 6, no. 8, pp ,. [7] J. Wu, G. Wang, an Y. R. Zheng, Energy efficiency an spectral efficiency traeoff in type-i ARQ systes, IEEE J. Sel. A. in Coun., vol. 3, no., pp , 4. [8] A. Mahoo an R. Jäntti, Packet error rate analysis of uncoe schees in block-faing channels using extree value theory, IEEE Coun. Lett., vol., no., pp. 8, Jan 7. [9] S. Cui, A. J. Golsith, an A. Bahai, Energy-constraine oulation optiization, IEEE Trans. W. Coun., vol. 4, no. 5, pp , 5. [] T. Wang, W. Heinzelan, an A. Seyei, Miniization of transceiver energy consuption in wireless sensor networks with AWGN channels, in 46th Conf. on Coun., Control, an Coputing, 8, pp [] Y. Hou, M. Haaura, an S. Zhang, Perforance traeoff with aaptive frae length an oulation in wireless network, in 5th Intl. Conf. on Coputer an Inf. Tech. IEEE, 5, pp [] M. Wu et al., New exponential lower bouns on the Gaussian Q- function via Jensen s inequality, in IEEE Veh. Tech. Conf.,.