Selection Relaying at Low Signal to Noise Ratios

Selecti on Relaying at Lo w Signal to Noise Ratios Ketan Rajawat and Ad rish Bane rjee Department o f Elec trical Eng ineering Indian Ins titute of T echnolog y Kanpur Kanpur-208016, India Email: ad rish@iitk.ac.in Abstract — Perf ormance of cooperative diversity schemes at Low Signal to Noise Ratios (LSNR) was recently studied by A v estimehr et. al. [1] who emphasized the importance of diversity gain ov er multiplexing gain at low SNRs. It has also been pointed out that continuous energy transfer to the channel is necessary fo r achieving the ma x-flow min-cut bound at LSNR. Motiv ated by this we propose the use of Selection Decode and Forward (SDF) at LSNR and analyze its perfo rmance in terms of the outage probability . W e also p ropose an energy optimization scheme which further brings do wn the outage pro bability . I . I N T RO D U C T I O N Cooperative di versity has attracted considerable intere st of researchers recently . The fo cus h as been o n d esign of efficient protoco ls, especially in the slow fading scenario [2], [3 ] where spatial diversity offers an inte resting metho d to combat the occasional deep f ades in th e channel. At high Sign al-to-Noise ratios (HSNR), th e perfor mance of a cooperative diversity protoco l is best measured by the div ersity-multip lexing trade of f it achie ves [3] , [4]. A t low SNRs howe ver , as shown in [1], the energy efficiency (and hence di versity) b ecomes f ar more im portant. Thus conventional schemes like Am plify and Forward (AF) and Decode an d Forward (DF) [2] become sub op timal at LSNRs because they are inefficient in the transfe r of energy to the netw ork. In [1], a novel Bursty AF (B AF) has been proposed that achieves full diversity at LSNRs. Th e max-flow min-cut bound was also der i ved and the perfor mance of B AF was shown to achieve th e bound u pto a first ord er ap proxim ation. When the so urce knows the source-relay c hannel gain, the perform ance can be further improved. This method was propo sed in [5 ] wh ere th e author s prop osed to switch between B AF and DF schemes b ased on the sou rce-relay chan nel. Usually the r equiremen t of CSI at transm itter in volves a f eed- back channe l which means extra r esources a nd more system complexity . Using the system model of [2] however , we see that f or a ny realistic system with two c ooperatin g mobiles, the mobiles must switch their ro les as source and relay . Thus the source-re lay ch annel inform ation acq uired while the mobile is acting as a relay can be used in the next block while acting as source thereby o bviating the need for feedback. W e also show that this channel inf ormation at the source allows us to use the s election relaying proto col. In a selectio n relaying protoco l like Selection Decode a nd Forward (SDF), the relay transmits only when it is able to dec ode the received signal completely (i.e. when the the source-r elay ch annel ex- ceeds a certain th reshold) otherwise th e source simp ly r epeats the transmission. Thus SDF also transmits energy continuously into the network thereb y ach ieving full di versity . I I . S Y S T E M M O D E L W e co nsider a system mod el similar to the on e sugg ested in [2], consisting o f two transmitting terminals an d on e recei ve terminal as sh own in Fig. 1. The no tation, baseband eq uiv alent s r d a rd a sr a sd Fig. 1. System Model model and the channel a llocation diagram is same as in [2 ]. Thus for direct tran smission, y d [ n ] = a sd x s [ n ] + z d [ n ] 1 ≤ n ≤ N / 2 (1) where N is the to tal block length (in nu mber of symbols) and x s [ n ] denote s th e signal transmitted by the source at time n . Th e no ise term z j for j ∈ { s, r , d } are also zero mean circularly symm etric comp lex Gaussian rando m variables with power spe ctral densities of N 0 . As in [ 2], the subscript is indicative of the r espectiv e terminal (source, relay or destina- tion). Similarly , for the other terminal, N/ 2 + 1 ≤ n ≤ N as shown in Fig. 2. F or the case of coope rativ e diversity , y d [ n ] = a sd x s [ n ] + z d [ n ] 1 ≤ n ≤ N / 4 (2) y r [ n ] = a sr x s [ n ] + z r [ n ] 1 ≤ n ≤ N / 4 (3) y d [ n ] = a r d x r [ n ] + z d [ n ] N / 4 + 1 ≤ n ≤ N / 2 (4) where a sr , a sd and a r d are the channe ls between sou rce- relay , source-d estination and relay-de stination respectively . Statistically th ese are modeled a s zer o mean, ind ependen t, circularly s ymmetric, complex Gaussian random v ariables w ith variances σ ij where i ∈ { s, r } and j ∈ { r, d } . As in [2] , the receiver is a ssumed to ha ve perfect knowledge of the chan nel T 2 Relay T 1 Tx + T 2 Rx N/2 N/4 T 1 Relay T 2 Tx + T 1 Rx T 1 Tx Direct Transmission Cooperative Diversity T 2 Tx Fig. 2. Channel A lloca tion for equal data case gain which remain con stant for N symbol inte rvals (i.e. time taken to transmit one blo ck). From now o n we will den ote g 1 = | a sd | 2 , g 2 = | a r d | 2 and h = | a sr | 2 Further, similar to [1] –[3], we assume similar imp lementa- tion constra ints, nam ely ha lf du plex chan nel, absence of CSI at tran smitter , and power co nstraints g iv en by ρ , P s N 0 (5) where P s is the power of each sym bol. Note that similar to [2], CSI abou t the sou rce-relay channel is s till a vail able to both sou rce an d relay as explain ed earlier . I I I . S E L E C T I O N D E C O D E A N D F O RW A R D I N L O W S N R The Selection Deco de and Forward (SDF) schem e was first propo sed in [2 ], wh ere its HSNR behavior was analyz ed. Here we ev aluate the perf ormance of SDF in LSNR an d show tha t it perf orms better than any of the existing schemes. The im portance of cooper ati ve diversity at LSNR was fir st analyzed in [1], where the auth ors sho wed that the impact of fading an d of di versity on the capacity , is much mo re significant at LSNR while multiplexing gain plays little role. They fu rther showed that th e Amplify and Forward (AF) and Decode and Forward (DF) sche mes perfo rm p oorly . In the AF scheme, the noisy signal received at the relay is amplified and re transmitted to the destination. Th is scheme fails at LSNR becau se th e relayed signal received at the destination is often too noisy to giv e di versity ad vantage. Based on this ob servation, the auth ors suggested the Bur sty AF (B AF) protoco l which by transmitting at low duty cycles and low rates actu ally overcomes this disadvantage. Another interesting scheme that was analyz ed was DF . It was shown that alth ough DF giv es f ull diversity , it do es n ot achieve th e max-flow min-cu t bo und. T he autho rs argued tha t achieving the bo und requ ires continuo us transfer of energy in the chan nel irr espectiv e o f th e chann el gain s. Th e DF sch eme, on the other hand, transmits in the second time slot only if it is ab le to co rrectly dec ode the inf ormation rece i ved fr om th e source in the first slot. The resulting discontinuity in energy transfer in e vitably redu ces the average energy transfered to the channel, resultin g in an increase in outage probab ility . T o overcome this p roblem of discontinuo us en ergy tr ansfer , we prop ose the use of SDF , where in the event of rela y being unable to deco de th e sou rce signal, the sou rce m ust retransmit its signal. Th is makes the transfer of energy continu ous an d as a result h elps achie ving better p erform ance. In fact for a higher o rder analysis, the SDF protoco l p erforms even better than the BAF . A similar ad aptiv e scheme ( AS) that switches between BAF and DF was proposed recently in [5 ]. The scheme utilizes source-re lay chann el gain at the receiver and perf orms b etter than the BAF scheme. A. P erformance at Low SNR A higher ord er analysis of v arious pro tocols a t LSNR was first done in [5 ]. T aking α = 2 R/ ρ for co nsistency with the system mode l, the re sults of [5] are repeated here for conv enience. Also fo r the sake of co mparison, we assume all channels to have unit variance (i.e. σ i = 1 ∀ i ). The outage probability of the B AF and AS pro tocols are giv en by P B AF = α 2 − 2 3 α 3 log α (6) P AS = α 2 + 2 α 3 (7) where similar to [1 ], α → 0 as ρ → 0 . Th e der i vation of above expression for BAF is very similar to the o ne o utlined in [1] excep t th at we con sider a highe r order term in all our appro ximations. W e now derive the p erform ance o f SDF scheme. Th e mutu al info rmation between th e sour ce and the destination (see [6]) was d erived in [2] an d is given by , I S D F = ( 1 2 log(1 + 2 ρg 1 ) 1 2 log(1 + hρ ) < R 1 2 log(1 + ρ [ g 1 + g 2 ]) 1 2 log(1 + hρ ) > R (8) where the factor of 2 app ears because of the rep etition cod ing of the source message. T he ou tage pro bability is now giv en by , P { I S D F < R } = P { g 1 < α/ 2 } P { h < α } + P { g 1 + g 2 < α } P { h > α } = α 2 2 + α 2 2 = α 2 (9) A higher order analysis can also be don e similarly an d gives, P S D F = α 2 − 29 24 α 3 (10) W e see that th e SDF per forms better th an the BAF protocol. In fact if we were to derive the expression for max- flow m in-cut bound upto higher ord er , it would turn out to be, P LB ≥ α 2 − α 3 (11) which is hig her than SDF . Th is is becau se this max-flow m in- cut b ound d erived in [1 ] assume d that the tr ansmitters h ad no CSI ( thus it assumed th e independe nce between the bits transmitted a t th e sou rce an d th e d estination, see [1 , App endix III] for d etails) which is no t th e case with the SDF pr otocol. For comparison we also sho w the outag e probabilities in Fig. 3 where the improvement provid ed by SDF over othe r schemes can ea sily be seen. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 0.005 0.01 0.015 α P out DF Bursty AF Adaptive Strategy Zero CSI Lower Bound Selection DF Fig. 3. Comparison of vari ous schemes at low SNR. α = 2 R/ρ I V . E N E R G Y O P T I M I Z A T I O N As h as alread y been po inted out in [1], en ergy is a treasured resource at LSNR. Thus energy optimization becomes far more importan t here. In [1] an energy op timization was done with respect to the amo unt of energy allocated to dif ferent slots for the m ax-flow min- cut bound. Howe ver as we h av e seen, the b ound has been achieved only a pproxim ately . W e should therefor e o ptimize the energy allocation with r espect to the outage pro bability itself rath er than the boun d. Assume that a fraction x of th e total power is dev oted to direct transmission and fractio n 1 − x to relayed (or dir ect as may be the case) transm ission. Thus th e SNR at destination is 2 xρ in fir st slot and 2(1 − x ) ρ in the second slot. The outa ge probab ility is now given by , P { I S D F < R } = P { g 1 < α/ 2 } P { 2 hx < α } + P { 2 g 1 x + 2 g 2 (1 − x ) < α } P { 2 hx > α } = α 2 4 x + α 2 8 x (1 − x ) = α 2 8  3 − 2 x x (1 − x )  (12) using results f rom [1]. M inimizing th e above expre ssion in x , we get x opt = 3 2 − √ 3 2 ≈ 0 . 634 (13) Notice that the correspond ing expression for max-flow m in-cut bound ( giv en in [1]) when ev aluated for symm etric case gives an optimu m value of x opt ≈ 0 . 667 which is slightly d ifferent from above. Similar d ifference is expected f or n on-symm etric case as well, particularly when asymm etry is large (ie. fo r the case when g 1 , g 2 and h differ considera bly). V . C O N C L U S I O N A N D F U T U R E W O R K The o ptimality of Selection Decode and Forward (SDF) was analy zed at low signal-to- noise r atio (LSNR) a nd shown to be better than the recently proposed bursty amplify and forward an d the adaptiv e schemes. Further we sho wed that energy optimization fo r SDF yield sligh tly different results from th at of max -flow min- cut boun d b ecause of the in herent sub optima lity . A P P E N D I X Here we der i ve the expression fo r the outage pro bability o f SDF scheme in term s of α using second o rder appro ximations. W e start with the general case assum ing variances of a sd , a r d and a a sr to be σ sd , σ r d and σ sr respectively . U sing the expression in (8), the outag e prob ability is given by , P { I S D F < R } = P { g 1 < e 2 R − 1 2 ρ } P { h < e 2 R − 1 ρ } + P { g 1 + g 2 < e 2 R − 1 ρ } P { h > e 2 R − 1 ρ } (14) Now g 1 , g 2 and h , as defined before, are exp onentially distributed random variables with means σ sd , σ r d and σ sr respectively . Th erefore g 1 + g 2 is exponentially distrib uted with its cumu lati ve distribution function g i ven by , P { g 1 + g 2 < x } = 1 − e − x (1 + x ) (15) Using these results in (14), we get P { I S D F < R } =  1 − e 1 − e 2 R 2 ρ   1 − e 1 − e 2 R ρ  +  1 − e − e 2 R − 1 ρ  1 + e 2 R − 1 ρ   e 1 − e 2 R ρ  (16) Since R , ρ an d R ρ approa ch ze ro, we may simply expand (16) using a T aylor series appro ximation and o btain , P { I S D F < R } =  2 R ρ  2 − 29 24  2 R ρ  3 + O [  2 R ρ  4 ] (17) where we h av e tak en R → 0 . No w setting α = 2 R /ρ as before, we o btain the desired result (10). R E F E R E N C E S [1] A. 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