Amplify-and-Forward in Wireless Relay Networks

Transcription

1 Amplify-and-Forward in Wirele Relay Nework Samar Agnihori, Sidharh Jaggi, and Minghua Chen Deparmen of Informaion Engineering, The Chinee Univeriy of Hong Kong, Hong Kong {amar, jaggi, Abrac A general cla of wirele relay nework wih a ingle ource-deinaion pair i conidered. Inermediae node in he nework employ an amplify-and-forward cheme o relay heir inpu ignal. In hi cae he overall inpu-oupu channel from he ource via he relay o he deinaion effecively behave a an inerymbol inerference channel wih colored noie. Unlike he previou work, we formulae he problem of he maximum achievable rae in hi eing a an opimizaion problem wih no aumpion on he nework ize, opology, and received ignal-o-noie raio. Previou work conidered only cenario wherein relay ue all heir power o amplify heir received ignal. We demonrae ha hi may no alway maximize he maximal achievable rae in amplify-and-forward relay nework. The propoed formulaion allow u o no only recover known reul on he performance of he amplify-and-forward cheme for ome imple relay nework bu alo characerize he performance of more complex amplify-and-forward relay nework which canno be addreed in a raighforward manner uing exiing approache. Uing cu-e argumen, we derive imple upper bound on he capaciy of general wirele relay nework. Through variou example, we how ha a large cla of amplify-and-forward relay nework can achieve rae wihin a conan facor of hee upper-bound aympoically in nework parameer. I. INTRODUCTION Since i inroducion in, Amplify-and-Forward (AF) relay cheme have been udied in he conex of cooperaive communicaion, 3, eimaing he capaciy of relay nework 4 6, and analog nework coding 7. For cooperaive communicaion, AF cheme provide paial diveriy o figh again fading; for capaciy eimaion of relay nework, uch cheme provide achievable lower bound ha are known o be opimal in ome communicaion cenario; and for analog nework coding, given he broadca naure of he wirele medium ha allow he mixing of he ignal in he air, hee cheme provide a communicaion raegy ha achieve high hroughpu wih low compuaional complexiy a inernal node. In hi paper, we concern ourelve moly wih he capaciy analyi of a general cla of Gauian AF relay nework. Exenion of our mehod and reul o cooperaive communicaion and analog nework coding cenario i he par of our fuure work. In he previou work, while analyzing he performance of AF cheme in relay nework one or more of he following aumpion have been made: nework wih a mall number of node 8, 0; nework wih imple opologie 3 5, 8, 0; or relay operaion in he high-snr regime, 0. However, for wo reaon, we believe ha i i imporan o characerize he performance of he AF cheme wihou uch aumpion. Fir, we feel ha for a cheme uch a amplify-and-forward ha allow u o exploi he broadca naure of he wirele medium uch aumpion on he ize and opology may reul in lower achievable performance han oherwie. Second, even in he low-snr regime amplify-andforward can be capaciy-achieving relay raegy in ome cenario, 5. Therefore, a framework o addre he performance of AF cheme in general wirele relay nework i deired. However, one major iue wih conrucing uch a framework i he following. In general wirele relay nework wih AF relaying, he reuling inpu-oupu channel beween he ource and he deinaion i an inerymbol inerference (ISI) channel (4, 0) wih colored noie. Thi i o becaue boh he ource ignal and he noie inroduced a he relay node may reach he deinaion via muliple pah wih differing delay. Thi reul in a formidable problem o analyze wih he exiing mehod wihou he aumpion above. Our main conribuion i ha we provide uch a framework o compue he maximum achievable rae wih AF cheme for a cla of general wirele relay nework, namely Gauian relay nework. Thi framework ca he problem of compuing he maximum rae achievable wih AF relay nework a an opimizaion problem. We emphaize ha our work how ha amplifying he received ignal o he maximum poible value a inermediae node migh reul in ub-opimal endo-end hroughpu. Alo, we eablih he generaliy of he propoed formulaion by howing ha i allow u o derive in a unified and imple manner no only he variou exiing reul on he performance of imple AF relay nework bu alo new reul for more complex nework ha canno be addreed in a raighforward manner wih exiing mehod. We how hrough variou example ha for a large cla of relay nework he AF cheme can achieve rae wihin a conan facor of he cu-e upper-bound on he capaciy of general wirele relay nework. In hi paper, we provide he ummary of our work. We have omied mo proof or give only brief ouline. The proof and dicuion can be found in our arxiv ubmiion 4. Organizaion: In Secion II we inroduce he general cla of Gauian AF relay nework addreed in hi paper. In Secion III we formulae he problem of maximum rae achievable via AF cheme in hee nework. In Secion IV we compue he rae achievable via AF cheme for wo inance of uch relay nework under variou communicaion cenario, and hen in Secion V we dicu he aympoic behavior of he gap beween hee rae and he correponding upper bound on he capaciy of general wirele relay nework compued here. Secion VI conclude he paper wih a ummary.

2 N M Relay N Fig.. A ingle ource-ingle deinaion communicaion channel over general Gauian relay nework wih M relay. II. SYSTEM MODEL Le u conider a (M )-node wirele relay nework wih ource deinaion and M relay a a direced graph G = (V, E) wih bidirecional link, a hown in Figure. Each node in he nework i aumed o have a ingle anenna. Le u aume ha he degree of he ource node i N, wih i being conneced o he deinaion node and a ube S of he relay node, N = S. Similarly, le u aume ha he degree of he deinaion node i N, wih i being conneced o he ource node and a ube S of he relay node, N = S. In general, S S V \{, }. A inan n, he channel oupu a node i, i V \{}, i y i n = h ji x j nz i n, < n <, () j N (i) where x j n i he channel inpu of he node j in he neighbor e N (i) of node i. In (), h ji i a real number repreening he channel gain along he link from relay j o relay i. I i aumed o be fixed (for example, a in a ingle realizaion of a fading proce) and known hroughou he nework. Furher, {z i n} i a equence (in n) of independenly and idenically diribued (i.i.d.) Gauian random variable wih zero mean and variance σ,z i n N (0,σ ). We alo aume ha z i are independen of he inpu ignal and of each oher. The ource ymbol x n, < n <, are i.i.d. Gauian random variable wih zero mean and variance P ha aifie average ource power conrain, x n N (0,P ). We aume ha he i h relay node ranmi power i conrained a: Ex i n P i, < n < () Moivaed by ome work on analog nework coding, uch a 5, le u modify he relay operaion in Secion II a follow. Relay operaion: We aume ha each relay node mainain a buffer of ignal i forwarded previouly. Therefore, each relay node i, afer receiving he channel oupu y i n a ime inan n execue he following erie of ep: Sep : Obain he reidual ignal y i n by ubracing he conribuion of previouly forwarded ignal from y i n. Sep : Compue he power P R,i of he reidual ignal y i n. Sep 3: A inan n ranmi he caled verion of he reidual ignal y i n of i inpu a ime inan n: x i n = β i y in, 0 β i β i,max = P i /P R,i, (3) where he β i i he caling facor. Le he nework-wide amplificaion vecor for he M relay node be denoed a β =(β,..., β M ). One of he major advanage of hi relay operaion i ha by ubracing he previouly forwarded ignal from heir inpu, he relay expend heir ranmi power in forwarding only he new informaion. Uing () and (3), he inpu-oupu channel beween he ource and deinaion can be wrien a an inerymbol inerference (ISI) channel ha a inan n i given by y n (4) = h x nz n D h i β i h ii... h id i d β id h id x n d d= D d=. D M d= (i,...,i d ) K d β h i... h id i d β id h id z n d (i,...,i d ) K,d (i,...,i d ) K M,d β M h Mi... h id i d β id h id z M n d, where K d, d D, i he e of d-uple of node indice correponding o all pah from he ource o he deinaion wih pah delay d and D i he delay of he longe uch pah. Noe ha along uch pah D M. Similarly, K m,d, m M, d D m, i he e of d-uple of node indice correponding o all pah from he m h relay o he deinaion wih pah delay d, D m i he delay of he longe uch pah from m h relay o he deinaion. I hould be noed ha max(d,..., D M )=D. Le u inroduce modified channel parameer a follow. For all he pah beween he ource and he deinaion: h 0 = h (5) h d = h i β i h ii... h id i d β id h id, d D (i,...,i d ) K d For all he pah beween he m h -relay, m M, and he deinaion: h m,0 =0, h m,d = (i,...,i d ) K m,d β m h mi... h id i d β id h id, d D m In erm of hee modified channel parameer, he ourcedeinaion ISI channel in (4) can be wrien a: y n = h j x n j h,j z n j (6) D... D M D h M,j z M n jz n Noe ha, in general, β i may depend on i h relay pa obervaion. β i n =f i,n (Y i n,..., Y i ). However, due o pracical conideraion, uch a low-delay operaion, we do no conider uch cenario here.

3 III. ACHIEVABLE RATE FOR THE SOURCE-DESTINATION ISI CHANNEL IN AF RELAY NETWORKS Lemma : For a given β, he achievable rae for he channel in (6) wih i.i.d. Gauian inpu i: I(P, β) = π log P H(λ) π 0 σ M m= H dλ, (7) m(λ) where H(λ) = h j e ijλ,h m (λ) = h m,j e ijλ,i= (8) D D m Proof: In a Dicree Fourier Tranform (DFT) baed formalim i developed o compue he capaciy of he Gauian channel wih ISI. We compue he maximum achievable rae for he channel in (6) for a given β by generalizing hi formalim o alo include he ISI channel for he Gauian noie a each relay node reuling in colored Gauian noie a he deinaion. The deail of he proof are in 4. For a given nework-wide amplificaion vecor β, he achievable informaion rae i given by I(P, β). Therefore, he maximum informaion-rae I AF (P ) achievable in an AF relay nework wih i.i.d. Gauian inpu i defined a he maximum of I(P, β) over all feaible β, ubjec o per relaynode amplificaion conrain (3). In oher word: (P): I AF (P ) def = max I(P, β) (9) β:0 βi β i,max The problem P implie ha in general AF relay nework, if he relay amplify he received ignal o he maximum poible hen i may reul in ub-opimal end-o-end hroughpu. I i differen from all he previou work where he relay ue all heir power o amplify he received ignal. The following example illurae he ignificance of hi obervaion. Example : Le u conider he well-udied diamond nework, 0. I i defined a G =(V, E) wih V = {,,, } and E = {(, ), (, ), (,), (,)}. Le h =,h = 0.,h = h =. Le P = P = P = 0 and noie variance σ =0. a each node. Therefore, we have β,max = P h P σ =0.99, P β,max = h P σ = 50.0 From (9), we have he following rae maximizaion problem: I AF = max β,β log 00 (β 0.β ) β β ubjec o conrain 0 β β,max and 0 β β,max. The objecive funcion i ploed in he Figure for β = The opimal oluion of hi problem i (β = 0.995,β = 0.5). Therefore, i follow ha in hi cae β = β,max i no he opimal amplificaion facor. Wih hi obervaion and he definiion of he relay operaion given in he previou ecion, i i appropriae o call he forwarding cheme propoed in hi paper a ubrac-caleand-forward. The problem P i compuaionally-hard o olve for all bu ome rivial relay nework 4. However under ome Achievable Rae β Fig.. The achievable rae for he Example when β = β,max and β lie in β,max,β,max. aumpion on relay operaion, i oluion can be efficienly approximaed. Thi we dicu in deail in 4 and demonrae below for wo inance of general AF relay nework. IV. APPROXIMATING THE RATE I AF (P ) Le u conider he problem P when he relay operae inananeouly, ha i he relay amplify-and-forward heir inpu ignal wihou delay, 8. Therefore, we have x i n = β i y i n, i M. Noe he poible yem inabiliy reuling from hi aumpion i avoided by he modified relay-operaion dicued above. Wih hi aumpion, (P) reduce o (afer eing λ =0in (9) and hen inegraing): (P): I AF (P ) = max β log P σ H(0) M m= H m(0) (0) uch ha 0 βi β i,max = P i/p R,i. Le u conider he problem P when he relay node are conrained o ue he ame amplificaion-facor, ha i, β i = β, for all i M. In he pracical eing, hi aumpion coniderably implifie he yem-deign wih β e o one paricular value for all relay node. Then, (P) reduce o (P3): I AF (P )= max log P H(0) 0 β βmax σ M H () m(0) Noe ha he oluion of (P) canno be maller han he oluion of (P3) becaue he e of feaible β : β i = β, for (P3) i a ube of he e of feaible β for (P). In general, βi,max P i. Therefore βi,max P i. However, a we are conidering he cae of equal β, β i,max = β max, o we have Mβmax P i or βmax M P i. Nex, we dicu he oluion of (P3) in differen cenario for wo pecial cae of he general cla of relay nework we addre in hi paper. A. Type-A Relay Nework Definiion: For one ource-deinaion pair and M relay, Type A nework i defined a: G = (V, E), where V = {,,,..., M} and E = {(, ), (, i), (i, ) : i {,..., M}}, a illuraed in Figure 3. We olve he problem P3 in he following wo cenario. Scenario (No aenuaion nework): Le u aume ha here i no aenuaion along any link in he nework, ha i,

4 h h M h M h h M h M h h h h M M Fig. 3. Type A Relay Nework. h = h i = h i =for all i M. The problem P3 in hi cae i: Propoiion : I AF (P ) = max 0 β βmax log P ( Mβ) σ Mβ Lemma : I AF (P ) aain global maximum a β op =. Now le u conider wo paricular way in which β max varie wih nework ize. Scenario, Cae A (Increaing relay power): For he given nework wih M relay node, le u aume ha he um power of he relay node i conrained a follow: M EXm m= M P i M u Q, u > 0,Q= conan m= So βmax = M u Q. From Lemma, we have for M. P log σ ( M), if β I AF (P )= max, () P log σ (M β max ), oherwie Scenario, Cae B (Conan oal relay power): Le u conider he cae where he um power of relay node i fixed irrepecive of he number of relay node, ha i M m= P i Q, Q = conan. Therefore, we e βmax = Q M. A M, for ufficienly large M, β max <β op =. Therefore, from Lemma, β = β max maximize he achievable rae and we have for M I AF (P )= log P σ Q Q M (3) Scenario (Bounded channel gain): Le u conider he cenario where he channel gain are arbirary, bu ricly bounded, 0 <h,h i,h i <, i M. The problem P3 in hi cae i: Propoiion : I AF (P ) = max 0 β βmax log P M (h β i= h ih i ) σ β M i= h i Lemma 3: I AF (P ) aain global maximum a β op = M i= hihi h M. i= h i Increaing relay power: Le u conider he increaing oal relay power cenario a in Scenario, Cae A. Le M h Fig. 4. h M,M Type B Relay Nework. βmax = M u Q. In hi cae, following Lemma 3, we obain he following lower bound on he achievable rae a M : I AF (P ) > log P σ (Mh min ) h,maxh min, (4) h max M i= if β max h ih i, M h i= h i I AF (P ) > log P σ (Mh min ) h,maxh min β max oherwie, M h max where h,max = max{h,h,..., h M }, h min = min{h,..., h M }, and h max = max{h,..., h M }. B. Type-B Relay Nework Definiion: For one ource-deinaion pair and M relay, Type B nework i defined a: G = (V, E), where V = {,,,..., M} and E = {(, ), (, i), (i, ), (j, j ) : i {,..., M},j {,..., M }}, a hown in Figure 4. For uch nework, we conider he no-aenuaion cenario where all channel gain are e o uniy, ha i, h = h i = h i =, i M a well a h i,i =, i M. The problem P3 in hi cae i: Propoiion 3: I AF (P,β) = ( log P βm β M β MβM β σ I AF (P ) = max 0 β β max, ) β βm (β ) ) M i= β( β M (i ) β I AF (P,β) (5) I can be proved ha he objecive funcion i quaiconcave 3, herefore a unique global maximum exi. Le β ha olve (5) be denoed a β = β op. However, obaining a cloed-form expreion for β op doe no appear raighforward, hough i can be numerically compued for any M. Conan oal relay power: Le u conider he cae where he um power of relay node i fixed irrepecive of he number of relay node a in Scenario, Cae B. Le βmax = Q M. A M, for ufficienly large M, β max < β op. Therefore, β = β max maximize he achievable rae and we have he following rae achievable aympoically a M I AF (P )= log P QM Q/M σ Q (6) Q/M

5 V. ASYMPTOTIC CAPACITY We fir derive an upper bound o he capaciy of general wirele relay nework we addre in hi paper. We hen dicu he aympoic behavior of he gap beween hi upper bound and he lower bound compued in he previou ecion. Propoiion 4: The capaciy C of a general wirele relay nework i upper-bounded a C min{c BC,C MAC }, where C BC and C MAC are he upper bound on he capaciy of he broadca cu a he ource and muliple-acce cu a he deinaion repecively, and are given a follow C BC = log P σ (h h i) i S C MAC = log P i S P i σ (h h i) i S Remark 3: The Propoiion 3 in 4 can be obained a a pecial cae of hi propoiion by eing N = N = M. A. Type A Relay Nework No aenuaion, increaing relay power: In hi cae, he broadca bound C BC i alway aympoically maller han he muliple-acce bound C MAC, a follow from C MAC C BC M u Q, for large M. P Therefore i uffice o compue he aympoic gap beween C BC and he lower bound in (). In fac, in hi cae we have C BC I AF (P )=0, for all M The acual capaciy C of he relay nework in hi cae i bounded by C BC / C C BC. No aenuaion, conan oal relay power: In hi cae alo he broadca bound C BC i aympoically maller han he muliple acce-bound C MAC, a hown below lim M C MAC C BC = Q P Therefore we only addre he aympoic gap beween C BC and he lower bound in (3). We have lim C BC I AF (P )= log( /Q) M The acual capaciy C of he relay nework in hi cae i bounded by (C BC log( /Q)) C C BC and he bound ge igher wih increaing Q. Bounded channel gain: The gap beween C BC and lower bound of achievable rae in (4) i bounded aympoically a: lim C BC I AF (P ) M log hmax h min Noe ha he apparen looene of he gap compued above compared o he correponding gap in 4 arie from he erie of implificaion made o reduce (P) o (P3). B. Type B Relay Nework No aenuaion, conan oal relay power: A claimed above, he upper-bound in Propoiion 4 hold for Type B nework oo. Therefore we only addre he aympoic gap beween C BC and he lower bound in (6). We have lim C BC I AF (P )= log( /Q) M The acual capaciy C of he relay nework in hi cae oo i bounded by (C BC log( /Q)) C C BC and bound ge igher wih increaing Q. VI. CONCLUSION AND FUTURE WORK We provide a framework o analyze he performance of he AF relay cheme in a general cla of Gauian relay nework. We how ha compared o he exiing mehod, he propoed framework no only allow u o characerize he performance of general AF relay nework in a unified manner bu i alo allow for igher characerizaion. We alo how ha AF cheme can be capaciy achieving for a large cla of wirele relay nework. An exenion of our work alo faciliae he compuaion of achievable rae for analog nework coding cenario for non-layered nework and low o moderae SNR regime. We plan o addre i in deail in our fuure work. REFERENCES J. N. Laneman, D. N. C. Te, and G. W. Wornell, Cooperaive diveriy in wirele nework: efficien proocol and ouage behavior, IEEE Tran. Inform. Theory, vol. IT-50, December 004. Y. Zhao, R. Adve, and T. J. Lim, Improving amplify-and-forward relay nework: opimal power allocaion veru elecion, IEEE Tran. Wirele. Comm., vol. TWC-6, Augu S. Borade, L. Zheng, and R. Gallager, Amplify-and-forward in wirele relay nework: Rae, diveriy, and nework ize, IEEE Tran. Inform. Theory, vol. IT-53, Ocober M. Gapar and M. Veerli, On he capaciy of large Gauian relay nework, IEEE Tran. Inform. Theory, vol. IT-5, March K. S. Gomadam and S. A. Jafar, Opimal relay funcionaliy for SNR maximizaion in memoryle relay nework, IEEE JSAC, vol. 5, February T. Cui, T. Ho, and J. Kliewer, Memoryle relay raegie for wo-way relay channel, IEEE Tran. Comm., vol. 57, Ocober S. Zhang, S. C. Liew, and P. P. Lam, Phyical-Layer Nework Coding, Proc. ACM MobiCom, Lo Angele, CA, Sepember S. Kai, I. Marić, A. Goldmih, D. Kaabi, and M. Médard, Join relaying and nework coding in wirele nework, Proc. IEEE ISIT 007, Nice, France, June S. Kai, S. Gollakoa, and D. Kaabi, Embracing wirele inerference: analog nework coding, Proc. ACM SIGCOMM, Kyoo, Japan, Augu, I. Marić, A. Goldmih, and M. Médard, Analog nework coding in he high-snr regime, Proc. IEEE WiNC 00, Boon, MA, June 00. A. Argyriou and A. Pandharipande, Cooperaive proocol for analog nework coding in diribued wirele nework, IEEE Tran. Wirele. Comm., vol. TWC-9, Ocober 00. W. Hir and J. L. Maey, Capaciy of he dicree-ime Gauian channel wih inerymbol inerference, IEEE Tran. Inform. Theory, vol. IT-34, May M. Avriel, W. E. Diewer, S. Schaible, and I. Zang, Generalized Concaviy. Plenum Pre, S. Agnihori, S. Jaggi, and M. Chen, Amplify-and-Forward in Wirele Relay Nework, arxiv:c.it:6 May 0. 5 Q. You, Z. Chen, Y. Li, and B. Vuceic, Muli-hop bi-direcional relay ranmiion cheme uing amplify-and-forward and analog nework coding, To appear in Proc. IEEE ICC 0, Kyoo, Japan, June 0.