Upper and Lower Bound on Signal-to-Noise Ratio Gains for Cooperative Relay Networks

Transcription

1 1 Upper and Lower Bound on Sgnal-to-Nose ato Gans for Cooperatve elay Networks Tobas enk, Fredrch K. Jondral Insttut für Nachrchtentechnk, Unverstät Karlsruhe TH Wreless Systems Lab, Stanford Unversty emal: Abstract Cooperatve networkng as a means of creatng spatal dversty s used n order to mtgate the adverse effect of fadng n a wreless channel and ncrease relablty of communcatons. We nvestgate sgnal-to-nose rato SN gan n wreless cooperatve networks. We show that the dfferental SN gan n the hgh data rate regme, whch we refer to as SN gan exponent ζ, s ndependent of the relayng strategy and only depends on the number of transmsson phases used for communcaton. Furthermore, a straght-lne upper and lower bound s derved based on geometrc consderatons. It s shown that the approxmaton error of the upper bound wth respect to the exact SN gan tends to zero for.for the lower bound, the approxmaton error tends asymptotcally to a constant factor δ for. Both bounds are the best possble straght-lne bounds wth respect to absolute error. Keywords cooperatve communcatons, relayng, sgnal-tonose rato gan, upper bound, lower bound. I. INTODUCTION The randomness of the wreless channel leads to tmevarant fluctuatons of the receved sgnal ampltude. Ths adverse effect of fadng can be mtgated by dversty technques [1], [2]. The creaton of spatal dversty s one of the benefts of relayng and cooperaton. In cooperatve networks moble nodes pool ther resources n order to acheve a better performance,.e., ncrease the relablty of communcatons. Along wth those benefts come challenges for the desgn of large scale dstrbuted networks such as cooperatve relay networks. Those challenges can be settled n the feld of network nformaton theory e.g., network codng and communcatons theory e.g., medum access, combnng recever structures. There are several performance metrcs for the evaluaton of cooperatve networks. Among the most mportant ones are achevable rate, dversty-multplexng tradeoff DMT [3], codng gan [4], and sgnal-to-nose rato SN gan [5]. In ths paper, we focus on the SN gan of cooperatve networks. We frst focus on the straght-lne upper bound and wll then use found results n order to derve the straght-lne lower bound. Indeed, t wll be shown that the lower bound s not very convenent, but t s nonetheless the best lnear lower bound possble. The paper s structured as follows. In Secton II we brefly summarze our man contrbutons. Secton III ntroduces our system model. We wll see that an abstract model s suffcent for the bass of our nvestgatons. A detaled analyss on SN gan s gven n Secton IV, whereas Secton V deals wth the dervaton of the upper and lower bound. We gve some numercal examples of very smple and well-understood cooperatve schemes to show the tghtness of our upper bound n Secton VI before we conclude the paper wth Secton VII. II. MAIN CONTIBUTIONS We establsh a unfed framework n order to evaluate cooperatve networks wth respect to SN gan. Noteworthy, that statements n ths paper are valed for 1 bt/s/hz whch seems reasonable from a practcal perspectve. We ntroduce a new performance crteron named SN gan exponent ζ whch descrbes the slope of SN gan for large values of and thus the dfferental behavor of SN gan. Ths new crteron s hence comparable to DMT ntroduced by Zheng and Tse n [3]. We show that n the hgh rate regme,.e., for, the SN gan exponent s only determned by the number of transmsson phases L and ndependent of the requred outage probablty or channel condtons. SN gan exponent can then be approxmated by ζ 31 L. Furthermore, we derve a straght-lne upper and a straghtlne lower bound on SN gan. It s shown that these two bounds are the best possble straght-lne bounds wth respect to absolute error. III. SYSTEM MODEL The channel gan between node and j s denoted by h j. A vector h contans all channel gans between nodes n the network. For nstance, consder a network wth one source, one relay, and one destnaton. Then, h becomes h sd,h sr,h rd. We consder an abstract channel model where we do not have to defne a specal probablty dstrbuton for the channel gans. Of course, when t comes to a more precse nvestgaton, then a proper probablty dstrbuton must be chosen n order to descrbe channel propertes. For nstance, when we consder a quas-statc transmsson scenaro where the delay requrements are short compared to the coherence tme of the channel, h j s modeled as a complex-valued crcular-symmetrc Gaussan random varable. Therefore, the magntude h j follows a aylegh dstrbuton. The channel nput of each node s lmted to average power P. We assume whte Gaussan nose to be added on each transmsson path whch follows CN0,N. SN s then gven by SN = P/N. We now defne some basc notatons that are used throughout the paper. s the target rate n bt/s/hz whch equals 2009 IEEE. Personal use of ths materal s permtted. Permsson from IEEE must be obtaned for all other uses, ncludng reprntng/republshng ths materal for advertsng or promotonal purposes, creatng new collectve works for resale or redstrbuton to servers or lsts, or reuse of any copyrghted component of ths work n other works.

2 2 bt/channel use. SN gan parametrzed on outage probablty s gven by Δp out. If we express SN gan n db, we denote t by Δ SN p out. The varable L descrbes the number of transmsson phases used for transmsson. Frst order dervaton of a functon fx s denoted by f x and second order dervaton by f x. Allsums n ths paper stand for L 1 =0 f not stated otherwse. IV. SIGNAL-TO-NOISE ATIO GAIN A. Defnton In order to dentfy the benefts of varous cooperaton strateges upon drect transmsson, we ntroduce the SN gan. SN gan of a cooperaton scheme over drect transmsson that acheves the same outage probablty p out s defned as Δp out = SNDT, 1 SN where the superscrpt DT descrbes the SN of drect transmsson and the superscrpt stands for varous relay strateges. SN gan s a functon of data rate, channel gans h, and outage probablty p out. Wth respect to our defnton n 1, SN gan can generally be factorzed as where Δ, h,p out =Δ th fh gp out, 2 Δ th = SNDT th SN th = 2 2 L s the rato of the requred threshold SNs and thus ratedependent, fh contans propagaton condtons, and gp out as a functon of outage probablty p out s lnked to dversty order. Expressed n db we get 3 Δ SN p out =10 10 Δ, h,p out. 4 In the followng, for the sake of descrpton, we wll smply use the notatons Δ and Δ SN. B. Geometrc Constructon The dervaton of our upper and lower bound s based on a smple geometrc constructon. Let SN gan be a monotoncally decreasng and concave functon both wll be proved later n. A straght-lne upper bound can then be derved by constructng a straght-lne wth the same slope as the SN curve for t can easly be seen that the slope for s the maxmal slope of the SN gan under these assumptons. Now, let ths straght-lne come from nfnty. We approach the curve of SN gan untl our straght-lne les upon the curve of SN gan. In order to derve the lower bound, we let the straght-lne come from mnus nfnty untl we touch the SN curve n the pont Q = =1bt/s/Hz, Δ SN. The prncple of the geometrc constructon s depcted n Fg. 1. We see that once we know the upper bound and the value for = 1 bt/s/hz, constructon of the lower bound s straghtforward. Fg. 1. Q ΔSN [db] upper bound lower bound [bt/s/hz] Geometrc constructon of upper and lower bound. C. Slope As already stated, the geometrc constructon of the upper and lower bound requre that SN gan s a monotoncally decreasng and concave functon n. We state the followng lemma. Lemma 1: SN gan Δ SN s a monotoncally decreasng functon n, that s: Δ SN = d d Proof: Snce fh and gp out are ndependent of rate and applyng the product rule for arthms to 2, we get Δ SN = d 2 d 10 2 L. 6 The factor 10 can be gnored as we are only nterested n the sgn of the dervatve. The dervatve can be drectly calculated from 6, however, as we wll see later, t s more convenent to express the argument by means of a geometrc seres. We have Δ SN = d d 10 = 10 e 2 = d 2 8 d Snce all summands of 9 are postve, Δ SN 0 and Lemma 1 s proved. A specal case occurs when SN gan s ndependent of. Ths s the case for ncremental relayng as ntroduced n [6] and transmt dversty [7]. The constructon of the upper and lower bound requres knowledge of the slope for. Ths s also of partcular nterest as t gves nformaton about the dfferental behavor 2009 IEEE. Personal use of ths materal s permtted. Permsson from IEEE must be obtaned for all other uses, ncludng reprntng/republshng ths materal for advertsng or promotonal purposes, creatng new collectve works for resale or redstrbuton to servers or lsts, or reuse of any copyrghted component of ths work n other works.

3 3 of varous cooperaton protocols. For nstance, for a specfc target rate, a cooperaton strategy that conssts of two relays mght be preferable to a one-relay protocol. However, ths may not be true for all values of dependent on the chosen cooperaton strategy. Therefore, t s advantageous to know the asymptotc behavor of SN gan for varous cooperaton strateges n order to fnd the most sutable one. From Fg. 1 we can conclude that Δ SN ζ,ζ 0, 10 where we defne the SN gan exponent as ζ =3 lm 2 Δ, h,p out [db/bt/s/hz]. 11 The factor 3 s due to the manpulaton of 10 nto 2 whch s preferable snce we deal wth expressons on mutual nformaton. Theorem 1: SN gan exponent ζ s only a functon of the number of transmsson phases L and can be approxmated by ζ 31 L. 12 Proof: A comparson of 11 and 12 ndcates that we have to show lm 2 Δ, h,p out =1 L. 13 We start wth the general representaton of Δ, h,p out n 2. Inserted n 11, we get lm 2 Δ th fh gp out = lm 2 Δ th snce fh and gp out are not rate-dependent and correspondng terms vansh for. We see that the SN gan exponent only depends on the rato of the requred threshold SN values. Ths yelds lm 2 Δ th = lm 2 2 / 2 L = 1 L whch concludes the proof. SN gan exponent ζ [db] depends lnearly on the number of transmsson phases L. Thus, we can conclude that the advantage n SN gan of mult-phase transmsson schemes s domnant for low-rate systems. Ths shows the tradeoff between the ablty to transmt wth a hgher data rate and the relablty of communcatons. D. Concavty Lemma 2: SN gan Δ SN s a concave functon n, that s: Δ SN = d2 d Proof: In 9 the frst dervatve of Δ SN s gven. Dervatng agan by applyng quotent rule and neglectng postve constants whch do not nfluence the sgn yelds SN gan Δ SN s concave f Δ SN 0, whch means that the nomnator of the above equaton satsfes After some manpulaton we get Ths s exactly the Cauchy-Schwarz nequalty a T ab T b a T b 2 wth a = 2,b = 2, and thus a b = 2. Ths proves Lemma 2. emark 1: An equvalent way to prove that SN gan s a concave functon and that avods the second dervatve s to show that the frst dervatve s a decreasng functon. Let ρ be an arbtrary number greater than or equal to 0 ρ 0. Then the followng nequalty must be satsfed: d d Δ SN Wth 9 ths becomes 2+ρ 2+ρ + ρ! d d Δ SN 18! The nomnator on both sdes ncreases faster than the correspondng denomnator due to the factor. Hence, condton 18 s met and SN gan s a concave functon. Equalty s acheved for ρ = 0 and/or L = 1. The latter corresponds to drect transmsson. Ths s, however, mmedately clear snce SN gan s defned accordng to the SN of drect transmsson. If ρ s fxed, equalty s also acheved for. V. BOUNDS A. Upper Bound Theorem 2: Let ζ [db] be the SN gan exponent for large values of rate and be gven by 12. Then SN gan can be upper bounded wth a straght lne by Δ SN ζ + t + =Δ + SN, 20 where t + s a functon of channel realzatons and outage probablty and s gven by t + fh gp out. 21 Proof: Consderng the generalzed expresson of SN gan 2 and 4, we get Δ SN Δ th fh gp out. 22 It can easly be seen that the second summand on the rghthand sde s t +. The frst summand can be manpulated nto: Δ th 2 L = = L L 1 31 L 2009 IEEE. Personal use of ths materal s permtted. Permsson from IEEE must be obtaned for all other uses, ncludng reprntng/republshng ths materal for advertsng or promotonal purposes, creatng new collectve works for resale or redstrbuton to servers or lsts, or reuse of any copyrghted component of ths work n other works.

4 4 The second equalty follows from a seres representaton equal to 7. Comparng the last equaton to 12, Theorem 2 s proved. emark 2: It can easly be seen that ths upper bound s the best straght-lne upper bound wth respect to absolute error that can be acheved. Consder the thrd equaton above. In order to fnd our upper bound, we took the largest value of the sum n the argument of the arthm and gnored the rest. Takng another value nstead and gnorng all remanng summands would defntely lead to a straght-lne upper bound that s less tght. Another possblty s to take the largest value and add 1 to t. Ths wll defntely lead to a tghter bound. However, ths s not a straght-lne bound anymore. Hence, we see that our upper bound s really the tghtest straght-lne upper bound that can be found to SN gan. Our constructon clearly ponts out that the upper bound converges asymptotcally to the exact value of SN gan. Accordngly, lm Δ + SN Δ Δ SN + 0 lm Δ The pont Δ + SN =1determnes the most convenent straght lne upper bound to SN gan. It can easly be shown that Δ + SN = 1 B. Lower Bound fh gpout 2 L For the constructon of the lower bound, we have to use the same slope as for the upper bound. Snce SN gan s generally a concave functon, the lower bound cannot converge to the exact value of SN gan. There wll be a dfference δ for hgh values of rate. Ths yelds lm ΔSN Δ SN δ lm Δ Δ δ = const. 25 wth δ = δ. Theorem 3: Let ζ [db] be the SN gan exponent for large values of rate and be gven by 12. Then SN gan can be lower bounded wth a straght lne by Δ SN ζ + t = ζ + Δ SN =1=Δ SN, 26 where t s a functon of channel realzatons, outage probablty and number of transmsson phases and s gven by fh t gpout 21 2 L. 27 Equalty s acheved for =1bt/s/Hz. Proof: The statement on equalty follows drectly from 26. As already mentoned, the lower bound s constructed by creatng a straght lne that ncludes Δ SN =1and possesses the slope of SN gan for, namely ζ. Then, from 26, we can see that t =Δ SN =1 ζ. We next calculate the value of SN gan for =1bt/s/Hz and get fh gpout Δ SN = 1 2 fh gpout 2 L, 28 where the second equaton follows from the presentaton of the sum as a geometrc seres. Consequently, fh t gpout 2 L 31 L fh gpout 1 2 L 2 1 L fh gpout 21 2 L, 29 whch proves 27. Corollary 1: The value t can be expressed n terms of t + and L as t = t L. 30 Proof: The proof follows drectly by comparng 27 to 21. We wll show later that the dfference between t + and t s the maxmal error of both bounds. C. Approxmaton Error In ths subsecton we nvestgate the tghtness of our bounds. For that purpose, we apply the absolute error whch can be expressed as ɛ ± abs = Δ± SN Δ SN. 31 As stated before, the lmtng behavor of the upper bound s gven by lm ɛ+ abs =0 32 and for the lower bound we have lm ɛ abs = δ. 33 Due to the constructon of the upper and lower bound, t becomes mmedately clear that the dfference between these bounds denotes the maxmal error. Accordngly, max ɛ ± abs = δ =Δ+ SN Δ SN. 34 Insertng 20 and 26 n 34 and applyng Corollary 1, we get δ = t + t 21 2 L. 35 For two transmsson phases ths becomes δ 1, 76 db. Hence, δ = 21 2 L =1.5. We wll come back to ths n the next secton, where we gve some smple examples n order to show the tghtness of our bounds. Fg. 2 llustrates the absolute error of the upper and lower bound over rate. Due to the geometrcal constructon, both curves ntersect for the value ɛ ± abs = δ/ IEEE. Personal use of ths materal s permtted. Permsson from IEEE must be obtaned for all other uses, ncludng reprntng/republshng ths materal for advertsng or promotonal purposes, creatng new collectve works for resale or redstrbuton to servers or lsts, or reuse of any copyrghted component of ths work n other works.

5 5 2 δ AM ɛ ± abs 1 lower bound ΔSN [db] 10 5 M 0.5 upper bound 0 DT MH [bt/s/hz] [bt/s/hz] 5 Fg. 2. Absolute error of upper and lower bounds wth respect to real value. VI. NUMEICAL EXAMPLES Fg. 3. SN gans Δ SN for mult-hop MH, mult-route M, and adaptve mult-route AM relayng wth upper bounds for p out =10 3 and α =4. elay has been placed half-way between source and destnaton,.e., d sr = 0.5. Drect transmsson DT serves as reference base. We now deal wth very smple examples to show the tghtness of our bounds. We consder networks that consst of one source, one relay, and one destnaton. The relayng strateges are mult-hop, mult-route, and adaptve mult-route, where the relay only forwards source nformaton f t has been able to decode properly. We have two transmsson phases and thus L =2. The path loss model s smlar to the one used n [8],.e., σj 2 = d α j, where σ2 j s the mean value of h j 2, α s the path loss factor, and d j denotes the dstance between node and j. The source-destnaton dstance has been normalzed to 1 and d rd =1 d sr. For a deeper anaylss on the dervaton of the followng bounds, we refer the reader to [9]. For mult-hop relayng, we derve the followng upper bound: Δ MH SN d α sr +1 d sr α 36 The upper bound for mult-route relayng eventually becomes after some algebrac manpulaton Δ M SN 3 0α 10 d sr. 37 Note that n both cases the upper bound, and thus SN gan n general, are ndependent of outage probablty. Ths s, however, not the case for adaptve mult-route relayng. Here, the upper bound becomes Δ AM SN d α sr +1 d sr α p out , 38 2 whch shows a 10 p out behavor. All three cases wth ther exact SN gans and the upper bounds are llustrated n Fg. 3. The lower bounds have been omtted for the sake of presentaton. For smulatons, outage probablty was p out =10 3, the relay has been placed halfway between source and destnaton and the path loss factor was set to α =4. VII. CONCLUDING EMAKS In ths paper we nvestgated the SN gans of cooperatve communcatons over drect transmsson. We showed that the dfferental SN gan, whch we ntroduced as SN gan exponent ζ, s ndependent of the relayng strategy n the hgh data rate regme and only depends on the number of transmsson phases L. It can be approxmated by ζ 31 L. Therefore, mult-phase transmsson schemes are advantageous for low-rate systems. We further derved a straght-lne upper and lower bound whch are both the best straght-lne bounds that can be acheved wth respect to the absolute error to the exact curve. EFEENCES [1] J. Proaks, Dgtal Communcatons, 4th ed. New York: McGraw Hll, [2] A. Goldsmth, Wreless Communcatons. Cambrdge Unversty Press, [3] L. Zheng and D. Tse, Dversty and multplexng: A fundamental tradeoff n multple antenna channels, IEEE Transactons on Informaton Theory, vol. 49, pp , May [4] J. N. Laneman, Network codng gan of cooperatve dversty, IEEE Mltary Communcatons Conference, Monterey, Calforna, USA, October/November [5] L. Sankaranarayanan, G. Kramer, and N. B. Mandayam, Cooperatve dversty n wreless networks: A geometry-nclusve analyss, 43rd Allerton Conference on Communcaton, Control and Computng, Montcello, Illnos, USA, September [6] J. Laneman and G. Wornell, Dstrbuted space-tme-coded protocols for explotng cooperatve dversty n wreless networks, IEEE Transactons on Informaton Theory, vol. 49, no. 10, pp , [7] P. Herhold, E. Zmmermann, and G. Fettwes, Cooperatve mult-hop transmsson n wreless networks, Journal on Computer Networks, vol. 49, no. 3, pp , [8] J. Laneman, D. Tse, and G. Wornell, Cooperatve dversty n wreless networks: Effcent protocols and outage behavor, IEEE Transactons on Informaton Theory, vol. 50, no. 12, pp , December [9] T. enk, H. Jaekel, and F. Jondral, elay-netzwerke - Kooperatve Kommunkaton unter moblen Telnehmern, Telekommunkaton Aktuell, vol , 61. Jahrgang, August/September, pp IEEE. Personal use of ths materal s permtted. Permsson from IEEE must be obtaned for all other uses, ncludng reprntng/republshng ths materal for advertsng or promotonal purposes, creatng new collectve works for resale or redstrbuton to servers or lsts, or reuse of any copyrghted component of ths work n other works.