Forwarding Strategies for Gaussian Parallel-Relay Networks

Transcription

1 1 Forwarding Strategies or Gaussian arallelrelay Networks vana Maric Member, EEE and Roy D. Yates Member, EEE Abstract This paper investigates reliable and unreliable orwarding strategies in a parallelrelay network. We consider the problem that maximizes the achievable rate under the total power constraint that allows or the power allocation among the nodes. We approach this problem by solving its dual with the objective to communicate to the destination at rate using minimum transmitted power. Motivated by applications in sensor networks, we assume large bandwidth res allowing orthogonal transmissions at the nodes. n such a network, the energy cost per inormation bit [1] during the reliable orwarding is minimized by operating in the wideband regime. For the wideband decodeandorward (DF) strategy, we present the optimum parallelrelay solution by identiying the best choice o relay nodes and the optimum power allocation among them. We demonstrate that the data should be sent over a single relay route through one relay that is in the best position in the network. On the other hand, as observed in [], the beneit o unreliable ampliyandorward (AF) strategy diminishes in the wideband regime. We characterize the optimum bandwidth or AF that minimizes the total energy cost per inormation bit or our network model. We show that transmitting in the optimum bandwidth allows the network to operate in the linear regime where the achieved rate increases linearly with transmit power. We then identiy the best subset o AF relay nodes and characterize the optimum power allocation per dimension among relays, or a given power and bandwidth. Based on this analysis, we compare the energyeiciency o DF and AF in a onerelay network and show the regions where each strategy is optimal. ndex Terms Decodeandorward, ampliyandorward, optimum relay strategy, optimum bandwidth. NTRODUCTON We consider a Gaussian parallelrelay channel consisting o a single destination pair and relays that dedicate their res to relaying inormation or the. The capacity o this network is not known or any inite. For, the capacity was ound in [3] or the degraded version o this channel. The asymptotic capacity in the limit as the number o relays tends to ininity was ound in [4], [], by employing unreliable orwarding at the relays. Another strategy in which relays do not decode a message, but send the compressed received values to the destination, was considered in [3] and extended to a multiple relay channel in [6]. When the relays are close to the destination, this strategy achieves the antennaclustering capacity [6]. This work was supported by New Jersey Commission on Science and Technology and NSF grant NSF AN The authors are with Wireless Network normation aboratory (WN AB), Department o Electrical and Computer Engineering, Rutgers University, iscataway, NJ 884 USA ( ivanam@winlab.rutgers.edu ryates@winlab.rutgers.edu). This approach is in contrast to the strategy where the message is reliably decoded at the relays, enabling relays to reencode and orward the message to the destination. This strategy achieves the capacity o the degraded relay channel [3]. ts extension to multiple relays was presented in [7], [8]. When relays are close to the, DF strategy can achieve the capacity in a wireless relay network [8]. This paper investigates reliable and unreliable orwarding strategies in a parallelrelay network. Motivated by applications in sensor networks, we assume large bandwidth res allowing orthogonal transmissions at the nodes. While the minimum energy cost per inormation bit in a general relay network is unknown [9], in our model, a decodeandorward network will minimize the energy cost per inormation bit [1] by operating in the wideband regime. However, as already observed in [], the AF strategy should operate in a dierent regime. A disadvantage o the AF strategy is that the relay power is wasted ampliying the receiver noise at a relay. This eect worsens as the signaling bandwidth increases. n act, or a network operating in the wideband regime, there is no beneit rom relays employing the AF strategy. With this observation in mind, we investigate the perormance o the reliable and unreliable orwarding in a parallelrelay network. We take approach o [1] and consider two dual problems one that maximizes the achievable rate given the power constraints and the second that minimizes the total average power to support a given rate at the destination node. We present the optimum solution or the wideband DF strategy by identiying the best choice o relay nodes and the optimum power allocation among them. We demonstrate that the data should be sent over a single relay route through one relay that is in the best position in the network. We then characterize the optimum bandwidth or AF and show that transmitting in the optimum bandwidth allows the network to operate in the linear regime where the achieved rate increases linearly with transmit power. We then identiy the best subset o AF relay nodes and determine the optimum power allocation per dimension among relays, or a given power and bandwidth. Based on this analysis, we compare the energyeiciency o DF and AF in a onerelay network while allowing each strategy to transmit in its optimum bandwidth. We identiy the regions where each strategy perorms better.. SYSTEM MODE A Gaussian parallelrelay channel is shown in Figure 1. The channel consists o a single destination pair and

2 @ * * Q * } D Fig. 1. α1 αm αm YR1 + 1 relay 1 Z1 YRm + relay m Zm β β 1 Y RM M β M + relay M + WM Z M arallelrelay Channel m β m + W + W1 relays that dedicate their res to relaying inormation or the. Each transmission occurs during a transmission interval in an orthogonal AWGN channel o bandwidth as in [11]. We view each orthogonal channel as a discretetime Gaussian channel by representing a waveorm o duration as a vector in the space or suiciently large [1]. n each time slot, relay observes a noisy version o the input transmitted at the (1) + Wm where is a Gaussian random variable with mean zero and variance! " and is the channel gain between the and relay. Relay transmits signal. The destination receives noisy copies o all inputs, as shown in Figure 1. The received signal at the destination at each time slot in a discretetime channel is #$&%' )( () % where is a diagonal channel gain matrix with entries * +, /.1! '93:. Vector 677 <>= is the vector o channel inputs and ( is the Gaussian noise vector with independent components with variance?!. For convenience, we assume the units o power are such that. We consider two dierent transmission strategies at the DecodeandForward: The transmission rom the is reliably received at a relay. The relay decodes the message, reencodes it and transmits that message in an orthogonal channel. AmpliyandForward: The received signal at relay, BA AC is ampliied and orwarded in an orthogonal channel. For ampliication gain * DFE, relay transmits,, HG. OTMUM BANDWDTH J To ormulate the problem, we irst consider the achievable rates per dimension or a given power allocation K : M 76M power at relay. Y (3) =ON where is the power and is the For the given power and the rate at the, relay will be able to execute the DF strategy only i the rate can be Q communicated reliably rom the to relay with power. Thus it has to hold that "Y SRUT4VW (4) Z When constraint Q Q (4) is met or node, we say that the makes node reliable. et [ G denote the set o reliable relays, i.e., those relays that are made reliable by a transmission at power and rate, and can thus execute the G DF strategy. For given [, the parallelrelay network () is a Gaussian vector channel analogous to a multiple antenna system [13]. The achievable rate per dimension is determined by the ' maximum mutual # inormation between the channel input and output. For the DF strategy, it is given by \^]`_ G K RaTbVdc e hg4i"jlk^m8n RUT4Vdc o e () On the other hand, any relay that is assigned power can ampliyandorward the received signal. The ampliication gain * is chosen such that the transmit power at relay is and is ound rom (3) to be p o b767q We can again use the result or Gaussian vector channels to determine the achievable rate or the AF strategy K RUT4Vsr t u +v " uo (6) xwby (7) The ollowing observation motivates the ormulation o our problem. emma 1: For any given power allocation K, \ ]`_/G K`z is increasing in or the decodeand orward strategy. For the ampliyandorward strategy, there exists inite that maximizes K`z. emma 1 states the well known act that the energy cost per inormation bit or the DF strategy is maximized when the network operates in the wideband regime [14]. On the other hand, emma 1 shows that the optimum operating bandwidth or the AF strategy is inite. Consider the maximum achievable rate in bits/s at the destination node or the AF strategy } & ~K where \ i_ is given by (7). For large, RUU > ƒ RU bits/s (8) (9) which is the rate achieved in the wideband regime by direct transmission rom the. Thereore, when operating in the wideband regime, there is no beneit rom transmissions at relays that employ AF strategy. This behavior was previously observed in [] in a parallel Gaussian network with two relays. With this observation in mind, we consider the problem to maximize the achievable rate in the network under the given power constraint. Rather than considering the power

3 Q = D D t A 3 constraint imposed on each transmitter, we consider a less restrictive power constraint on the total power in the network <: ' ˆ' A&. Because DF and AF will not in general operate in the same bandwidth, we compare the perormance o two strategies, using the total power constraint N K AŠ (1) The problem o maximizing the achievable rate under the total power constraint (1) has its dual with the objective to communicate to the destination at rate bits/s using minimum transmitted power. The two problems have analogous solutions that become the same when Œ. n the rest o the paper we present the solution to the latter problem. V. DECODEANDFORWARD When transmitting with rate, the will make node reliable i constraint (4) is met. The achieved rate at the destination is then given by (). This implies that, or the to communicate at rate with the destination, we require ŒRUT4Vdc e Mg4i"jk7mŽn ŒRUT4Vdc o e (11) To optimize the transmit powers, we have to ind the best subset o nodes to be made reliable so that they can decodeandorward the message. We use binary variables to indicate which relays will be reliable and ormulate our problem in the ollowing way: U Uv o (1) subject to (13) SRUT4V c e av" ŒRUT4Vdc SRaTbV W " e Zs ` s!e 74 b o DCE (1a) (1b) (1c) (1d) n the limit o large, this problem simpliies to the wideband DF relay problem: U Uv subject to Uv D Ẽ 7z DCE (14) (14a) (14b) (14c) (14d) Theorem 1: The wideband DF relay problem (14) admits an optimal solution in which no more than one relay node transmits. Theorem 1 holds because when transmitting with severely restricted transmitter power, water illing over the relays results in transmitting to the relay with the best channel to the destination. By Theorem 1, it is suicient to consider only policies that employ a single relay š. n this case, the problem (14) becomes the wideband single relay problem U œ subject to ) žd "žd ž s!e 74 b M?DCE (1) (1a) (1b) (1c) (1d) n the problem (1), one can show that relay š is used only i. n this case, the transmit powers are and BŸ 6z The total transmitted power o this solution is " Ÿ " (16) (17) We emphasize that this is the optimal power assignment or using node š as long as node š is a useul relay, in the sense that š belong to the set o useul relays Q^ b (18) Finally, among all useul relays š, we choose that one which minimizes the total transmitted power. We summarize our observations in the ollowing theorem. Theorem : the set o useul relays is nonempty, the optimal solution to the wideband DF relay problem (14) is or the to employ relay š p z V~ a g Ÿ œ (19) with power assignment given by (16) otherwise, i is empty, then direct transmission rom the to the destination is optimal. V. AMFYANDFORWARD We now deine the power minimization problem or the AF strategy. We ormulate the problem in terms o the power per dimension K`z. The achievable rate (7) becomes «RUT4Vsr H +v " () Since the optimum bandwidth in this case is not known, we ormulate the AF relay problem as a N (1) subject to D (1a) DCE (1b) EA s ~ Ž± (1c) w«y

4 ³ ³ E ¹ ¹ E E š š ¹ ² G ² Ç t 4 We assume that s ~ Ž± is a maximum bandwidth available in each channel. We choose s ~ q± suiciently large to allow the network to operate in the wideband regime. et? denote the optimum power and bandwidth allocation that achieves in (1). We irst observe that, to achieve nonzero rate in (1) and thus satisy the rate constraint (1a), the network uses bandwidth and the transmits with power?. Furthermore, constraint (1a) is always binding. Depending on the values o the channel gains, a solution to problem (1) may be a direct transmission rom the, that is,? &E or all 4676q, ~ Ž± and? given by (1a). Otherwise, there is a subset o ² AC relays employing the AF strategy. The agrangian in roblem (1) is ³ +v Ÿ) G ŸŠ () We next characterize the optimum bandwidth or the AF strategy in (1). emma : For s ~ Ž± suiciently large, the optimum bandwidth or the AF strategy in (1) is strictly smaller than µ ~ q±. roo or emma ollows rom the act that, or any ixed K, K~4 is decreasing in or large. Given? or the AF strategy, it would be convenient to b776^ relabel the nodes such that ² relays are the active transmitters with powers and E 4767^ ². From emma, the optimum solution or the AF strategy in (1) is never on the boundary o constraint (1c), that is E implies ~ q±. By the KuhnTucker conditions, this º¹ +v Ÿ pe From (3), we obtain the agrange multiplier &E 677 x» \ i_¼g ¹ +v? (3) (4) For nodes š ² with nonzero transmitter powers, the KuhnTucker conditions are & Ÿ½ From (4) and (),? pe &Ẽ 776^??» ¹ +v pe 767 G The optimum power allocation? 7676? determined rom ²¾ () (6) can then be equations given by (6), independent o and. The optimum bandwidth can be determined such that the solution lies on the easibility region boundary (1a): z\^i_ G (7) where the G irst ² elements o? are nonzero and are given by?? 6776? and the rest ŸÀ²¾ elements are zero. From (1), (4) and (7), v» ¹ +v? (8) We thus proved the ollowing: Theorem 3: The AF relay problem (1) has an optimum solution in which the optimum bandwidth, rate and the total transmit power have a linear relationship. We can view as a power price : increasing the required rate in (1) by Áž, increases the minimum required total power by Áž. The optimum solution to the AF relay problem (1) is speciied by the optimum transmit powers at the and the relays, operating in the optimum bandwidth. To determine the optimum power allocation requires solving the system o equations given by (6) which does not appear as an easy task. We approach this problem by considering a subproblem o (1) that determines the optimum power allocation per dimension among relays only, or a given power and bandwidth. A. Optimum ower Allocation at Relays To identiy the best subset o AF relay nodes or any power and signaling bandwidth, we let,  à " The achievable rate per dimension () becomes RUT4V r H +v ÂœÄ " Â Ä w«y (9) (3) For ixed bandwidth and power at the, maximizing over the choice o relay powers is equivalent to È: zå Æ +v subject to!7676 " ÂœÄ Â Ä +v A Ç (31) DCE (31a) (31b) where <= is the vector o relay transmit powers. The power per dimension allocated to relays is ½ Ÿ½ (3) where is the total power constraint as given by (1). Note that (31) solves the dual to (1) and or ½ the two problems have identical solutions. From the KuhnTucker conditions, the optimum solution to (31) is " Â É Ê Ÿ  ÌË b767 ² (33) Ê where is the agrange multiplier and can be ound such that boundary constraint (31a) G is satisied with equality. When solving (31) or optimum?, rom (3) and (31a) it ollows that» +vàí Î z ŸÏ Ç, and thus solves (1). Alternatively, we could derive solution (33) directly rom (). From (33) we observe that the optimum AF strategy in general employs a set o relays, unlike the optimum DF strategy. The optimum choice o relays strongly depends on

5 x ower allocation y coordinate destination 1 x coordinate 3 ower allocation y coordinate destination 1 x coordinate 3 Fig.. ower allocation among relays when operating in low SNR at the Fig. 3. ower allocation among relays when operating in high SNR at the the transmit power. Figures and 3 show the optimum power allocation among the relays spread over an area in between and destination, or two signiicantly dierent powers. For a operating at lowsnr, Figure shows that the best choice is to relay through the nodes that are close to the. This is not surprising since, in this case, the broadcast part o the channel rom the to relays is weak. The solution thus tries to maximize the perormance o the broadcast side by choosing nodes that have the highest received SNR. The situation is opposite in the high SNR regime at the, shown in Figure 3, where the SNR at the relays is high and relaydestination channel limits the perormance o the network. The solution then choses relays that have better channel to the destination. Thus, the best choice o relay nodes signiicantly depends on the transmit power at the and our next goal is to solve (1) and determine the optimum operating point. For two limiting power regimes, it ollows rom (9) that the parameter  is determined by the channel gains as  ÑÐ ž, z in lowsnr regime in highsnr regime (34) and is thus independent rom when is operating in lowsnr regime. On the other hand, rom (33), a node will be select as a relay when  Ê, or any given choice o bandwidth and Ê power. ncreasing will have an eect o increasing and thus o shutting down relays in the increasing order o Â. The similar eect will occur when is operating in highsnr regime. This observation can then be used to perorm the search or the optimum power. Finally, we compare the energyeiciency o DF and AF supporting the ixed rate in the onerelay network. For DF, the network operates in the wideband regime. The total power or DF is given by (17). For AF, or a given bandwidth, the optimum power allocation per dimension is determined rom (1) and the optimum bandwidth is ound numerically. The comparison o the powereiciency is shown in Figure 4. y coordinate Decode and Forward and Ampliy and Forward x coordinate Fig. 4. Decodeandorward and ampliyandorward perormance. The dots, circles and pluses indicate, respectively, the regions where DF, AF and direct transmission support the rate with minimum power. REFERENCES [1] S. Verdú, On channel capacity per unit cost, EEE Trans. on normation Theory, vol. 36, no., pp , Sept [] B. E. Schein, Distributed coordination in network inormation theory, in h.d thesis, Massachusetts nstitute o Technology, Sept. 1. [3] T. Cover and A. E. Gamal, Capacity theorems or the relay channel, EEE Trans. on normation Theory, vol., no., pp. 7 84, Sept [4] M. Gastpar, To code or not to code, in h.d. Thesis, Swiss Federal nstitute o Technology (EF), ausanne, Switzerland, Nov.. [] M. Gastpar and M. Vetterli, On asymptotic capacity o gaussian relay networks, in roc. o nternational Symposium on normation Theory (ST ), June. [6] M. Gastpar, G. Kramer, and. Gupta, The multiplerelay channel: Coding and antennaclustering capacity, in roc. o nternational Symposium on normation Theory (ST ), June. [7]. Gupta and. R. Kumar, Towards an inormation theory o large networks: An achievable rate region, EEE Trans. on normation Theory, vol. 49, pp , Aug. 3. [8] G. Kramer, M. Gastpar, and. Gupta, Capacity theorems or wireless dest

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