Resource Allocation for Cooperative Transmission in Wireless Networks with Orthogonal Users

Transcription

1 Resource Allocation for Cooperative Transmission in Wireless Networks with Orthogonal Users D. Richar Brown III Electrical an Computer Engineering Department Worcester Polytechnic Institute Worcester, MA 19 Abstract This paper investigates the problem of efficient power allocation in a wireless communication system with two cooperating sources an one estination. The sources in the system each transmit information to a single estination at a fixe SNR target an cooperate via an orthogonal amplify-anforwar protocol with two timeslots. We evelop a framework for power allocation in this scenario aroun the concept of cooperation ratios an erive expressions for the transmit power require by each source to achieve their SNR targets as a function of these cooperation ratios. Numerical examples are presente for time-invariant channels as well as Rayleigh faing channels. Our results show that cooperation oes not reuce require transmit powers when both sources have symmetric time-invariant channels to the estination. When sources have asymmetric time-invariant channels to the estination, total power is minimize when only the source with the stronger channel cooperates. In the case of Rayleigh faing channels, we emonstrate that mutual cooperation can minimize the average total require transmit power an can also lea to a reuction in average require transmit power for both sources. I. INTRODUCTION Recently, researchers have recognize that spatial iversity can be achieve in multiuser communication systems even if the noes in the system each have only one antenna. Senonaris, Erkip, an Aazhang were the first to propose the concept of user cooperation iversity where nearby users in a cellular system form cooperative partnerships by sharing their antennas to achieve increase rate or ecrease outage probability in the uplink [1]. Since this seminal work, there has been a growing interest in eveloping cooperative transmission protocols an unerstaning the performance limits of user cooperation iversity, c.f. [] []. In aition to its emonstrate potential for increase rate or ecrease outage probability, the user cooperation iversity can also potentially reuce the transmit power require by noes to meet QoS targets an, consequently, exten the battery life of cooperating noes [1]. This may be particularly important in energy-constraine scenarios such as sensor networks. Cooperative transmission is unique, however, in that it requires autonomous noes to allocate transmit power between selfish an cooperative transmissions. Inefficient allocation of transmit power coul lea to worse power efficiency than no cooperation. The primary focus of this paper is on the problem of how to allocate transmit power between selfish an cooperative transmissions in orer to maximize power efficiency in a wireless communication system with two cooperating noes communicating inepenent information over orthogonal subchannels to one estination. Our analysis also consiers the problem of fairness: Uner what conitions o both noes benefit (in terms of reuce transmit power) from cooperation? This question may be important towars eveloping a better unerstaning of the problem of inucing cooperation between autonomous noes. II. SYSTEM MODEL AND COOPERATIVE PROTOCOL The two-source cooperative transmission system moel consiere in this paper is shown in Figure 1. The channels are assume to be flat an either time-invariant or block-faing where their value is ranomly generate but remains constant over both timeslots of the cooperative protocol escribe below. The channels are also assume to be known to both sources as well as the estination. Fig. 1. S 1 h 1 h 1 S g 1 Two-source cooperative transmission system moel. The two-source amplify-an-forwar [7] cooperative transmission protocol is given in Table I. Transmission occurs in two timeslots: each source transmits its own information in the first timeslot; the secon timeslot is use for cooperative retransmission. We enote the i th source s zero-mean unitvariance information symbol as x i, the i th source s amplitue in the k th timeslot as a i [k], an the i th source s transmission in the k th timeslot as t i [k]. The j th source in the system receives the signal sent by the i th source in timeslot k as g r ij [k] =h ij t i [k]+v ij [k] where h ij is the (scalar) channel gain in orthogonal channel i between source i an source j an v ij [k] is the zero-mean noise in this channel with variance σ v >. D

2 TABLE I TWO-SOURCE AMPLIFY-AND-FORWARD COOPERATIVE PROTOCOL. TABLE II TRANSMIT POWERS BY SOURCE AND TIMESLOT. timeslot 1 timeslot Source 1 t 1 [1] = a 1 [1]x 1 t 1 [] = a 1 []r 1 [1] Source t [1] = a [1]x t [] = a []r 1 [1] timeslot 1 timeslot Source 1 P 1 [1] = a 1 [1] P 1[] = a 1 [] `h 1 a [1] + σ v Source P [1] = a [1] P [] = a [] `h 1 a 1 [1] + σ v The signal receive by the estination in the i th orthogonal channel in timeslot k is given as y i [k] =g i t i [k]+w i [k] where g i is the scalar channel gain between the i th user an the estination an w i [k] is the zero-mean noise in this channel with variance >. The estination forms the ecision statistic for the i th user s information symbol as a linear combination of the two relevant observations, i.e., y 1 = b 1 [1]y 1 [1] + b []y [] y = b [1]y [1] + b 1 []y 1 [] where b i [k] are the linear combination parameters selecte by the estination to maximize the SNR of the ecision statistics. Note that there is no multiaccess interference ue to the orthogonality of all transmissions in this protocol. III. TRANSMIT POWER AND SNR ANALYSIS Given the system moel an cooperative transmit protocol escribe in Section II, we consier the problem of efficiently allocating transmit power in orer to achieve a pair of fixe SNR targets, enote as SNR 1 an SNR, at the estination. In the absence of cooperation, the orthogonality of the sources makes the solution to this problem straightforwar. The SNR targets will be satisfie iff P i = a i [1] (σ w/gi )SNR i where P i enotes the transmit power of the i th source. The problem of power allocation in the cooperative scenario is less straightforwar, however, ue to the fact that there are four transmit powers to specify with only two constraints. This section evelops an analytical framework for power allocation in the two-source cooperative transmission scenario in orer to better unerstan how cooperation influences the iniviual power requirements of each source as well as the total power requirement for both sources. A. Transmit Powers an Cooperation Ratios The first step in our analysis is to calculate the transmit power for each source in the original an cooperative timeslots. Denoting the transmit power of source i in timeslot k as P i [k] := E[t i [k]], the transmit powers in each timeslot (conitione on the channel realizations) are given explicitly in Table II. The total transmit power for source i is given as P i = P i [1]+P i [] an the total transmit power over all sources is given as P tot = P 1 + P. From the power expressions in Table II, we can efine a cooperation ratio parameter for each source in the system. The i th source s cooperation ratio is efine as the ratio of the power of the i th source s cooperative retransmission to the power of the original transmission of source j (i j). Using the results from Table II, we can write the cooperation ratios for the i th source as α i := P i[] P j [1] = a i [] ( h ji a j [1] + v) σ a j [1] i, j {1, } for α i < an j i. We note that the noncooperative case correspons to α 1 = α =. B. Destination Processing an SNR The next step in our analysis is to erive expressions for the SNR of y 1 an y at the estination uner the assumption that the estination optimally combines the observations {y 1 [1],y 1 [],y [1],y []} to maximize SNR. Assuming that all of the noise terms in the observations are mutually inepenent as well as inepenent of the ata, it can be shown that maximal ratio combining (MRC) at the estination maximizes the SNR of y 1 an y. The MRC combining coefficients can be written as b [] g h 1 a [] = b 1 [1] g 1 ( + g a []σ v ) b 1 [] g 1 h 1 a 1 [] = b [1] g ( + g1 a 1 []σ v) an the resulting SNRs with MRC at the estination can be expresse as SNR 1 = g 1 a 1 [1] SNR = g a [1] + g a []h 1 a 1 [1] σ w + g a []σ v (1) + g 1 a 1 []h 1 a [1] +. () g 1 a 1 []σ v Substituting the cooperation ratios an the transmit powers from Table II into these expressions, we can rewrite the SNR of each source as SNR 1 = g 1 P 1[1] α P1 + [1]g h 1 (h 1 P 1[1] + σv )+α P 1 [1]g σ v SNR = g P [1] α 1 P [1]g + 1h 1 (h 1 P [1] + σv)+α 1 P [1]g1. σ v We note that specification of the SNR targets {SNR 1, SNR } as well as the cooperation ratios {α 1,α } fully etermines the minimum transmit powers for both sources in first timeslot an the resulting cooperative transmit powers in the secon timeslot.

3 IV. RESULTS This section uses the analysis of Section III to examine the transmit power require by each source as well as the total transmit power require to satisfy a pair of SNR targets SNR 1 an SNR at a fixe level of cooperation specifie by α 1 an α. We consier two scenarios istinguishe by the channel moel: time-invariant channels an Rayleigh faing channels. Numerical examples are presente in orer to evelop insight into the following questions: 1) What choice of cooperation ratios α 1 an α minimize the total require transmit power P tot? ) How can we escribe the set of mutually beneficial cooperation ratios A = { <α 1,α < P i (α 1,α ) < P i (, ) i {1, }}? Uner what conitions is this set empty or non-empty? All numerical results in this section have fixe SNR targets specifie as SNR 1 = SNR =1B. A. Time-Invariant Channels In this section, we examine the transmit power requirements for the case when all of the channels in Figure 1 are all moele as time-invariant. We first consier the case where g 1 = g an h 1 = h 1, i.e., symmetric channels. In this case, it can be shown that cooperative transmission can result in reuce iniviual transmit power for one source at the expense of increase transmit power for the other source. A simultaneous reuction in iniviual transmit powers P 1 an P with respect to the noncooperative case is not possible, however, hence A =. Moreover, the total require transmit power is minimize when α 1 = α =. In other wors, cooperation cannot provie a reuction in total transmit power with respect to the noncooperative case. These results can be intuitively explaine by the fact that, if all of the channels are symmetric, it is more effective to put power into the the first timeslot than to amplify an forwar the noisy signal from the other source in the secon timeslot. Figure shows the require transmit powers P 1, P, an P tot as a function of the cooperation ratios α 1 an α for a particular example of the symmetric time-invariant channel case. A more interesting case occurs when the source-estination channels g 1 an g are asymmetric. In this case, it can be shown that, like the symmetric case, cooperative transmission can reuce the iniviual transmit power for one source at the expense of increase transmit power for the other source. A simultaneous reuction in iniviual transmit powers is not possible an, again, A =. Unlike the symmetric case, however, as long as the source-source channel is better than the weaker source-estination channel, cooperation can reuce the total require transmit power with respect to the noncooperative case when the source with the stronger sourceestination channel cooperates. In this sense, mutual cooperation is unesirable: only the source with the stronger channel to the estination shoul retransmit in the secon timeslot. Figure 3 shows a particular example of the time-invariant asymmetric channel case when g 1 <g an h 1 = h 1. B. Faing Channels In this section, we analyze the average transmit power require to meet the fixe SNR targets {SNR 1, SNR } for a fixe set of cooperation ratios {α 1,α } in the case when all of the channels in Figure 1 are moele as flat inepenent Rayleigh faing. In this case, the sources allocate power accoring to the current channel state in orer to meet their SNR targets. We assume that there is no maximum power constraint on either source in this analysis. In orer to quantify the benefits of cooperation, we first consier the problem of meeting a fixe SNR target in the noncooperative case with Rayleigh faing channels. In this case, all power is allocate to timeslot 1 an the i th source s require transmit power, conitione on the channel realization, is P i =( /g i )SNR i. When g i is moele as a Rayeligh istribute ranom variable, it can be shown that gi is exponentially istribute an that the average transmit power E[P i ] is infinite irrespective of the mean of g i. Proposal 1: Given g 1, g, h 1, an h 1 are inepenent an Rayleigh istribute, an α 1,α (, ), the average transmit powers E[P 1 ] an E[P ] require to meet the fixe SNR targets SNR 1 < an SNR < are finite. Proof sketch: Due to space limitations, a sketch of the proof to Proposition 1 is provie here. The average total transmit power by the i th source can be expresse as E[P i ]=E[P i [1]]+ E[P i []].Byefinition of the cooperation ratio α i, we can state that E[P i []] < if α i is finite an E[P j [1]] < for j i. Hence, given the assumption that the cooperation ratios are finite, each source s total average require transmit power is finite if E[P 1 [1]] < an E[P [1]] <. Isolating the timeslot 1 transmit power for the i th source, we can rewrite (1) as P i [1] = a i [1] = SNR + a j []σ w σ v Y i X + a j []σ vxy + a j []σ wyz where we have substitute X = gi, Y = g j, an Z = h ij for the ranom inepenent exponentially istribute squarechannel coefficients. For notational convenience, we normalize the means of X, Y, an Z an collect the non-ranom parameters to write θ + θ 1 Y P i [1] = θ X + θ 3 XY + θ YZ = P i[1] + P i1 [1] where <θ i < for all i {,...,} an P i [1] an P i1 [1] correspon to the fraction forme by the first an secon terms of the numerator, respectively. To show that E[P i [1]] is finite, we will show separately that E[P i [1]] < an E[P i1 [1]] <. We can upper boun P i1 [1] by θ 1 Y P i1 [1] = θ X + θ 3 XY + θ YZ θ 1 θ 3 X + θ Z c X + Z for some finite constant c where we have use the fact that θ X. An upper boun on the expectation of P i1 [1] can then be written as E[P i1 [1]] c x + z e x e z x z

4 which can be shown to be finite by using the bouns e x (1 + x) 1 an e y (1 + y) 1 for all x, y. Following a similar proceure for P i [1], we can write an upper boun θ P i [1] = θ X + θ 3 XY + θ YZ θ θ X + θ YZ X + YZ for some finite constant where we have use the fact that θ 3 XY. An upper boun on the expectation of P i [1] can then be written as E[P i [1]] x + yz e x e y e z x y z. We can use the continuity of the integran an the monotonicity of the exponential terms to write a looser upper boun on this expectation as E[P i[1]] X X X Z j+1 e j e k e l j=k=l= j 1 1 Z k+1 Z l+1 k l x + yz x y z. It can be shown that 1 x+yz x y z is finite an, consequently, that E[P i [1]] is finite. Since both P i [1] an P i1 [1] are finite, P i [1] is finite an the total average require transmit power for both sources is finite. We note, as a technical etail, that the proof of Proposition 1 oes not require the source-source channels h 1 an h 1 to be inepenent (which may not be the case if these channels are reciprocal) but only that the sets of channels {g 1,g,h 1 } an {g 1,g,h 1 } are inepenent. Proposition 1 implies that any level of mutual cooperation is beneficial, in the sense of reucing the require average transmit power, when the channels are moele as inepenent Rayleigh faing since the average require transmit power in the noncooperative case is infinite for both sources. Hence A =(, ) (, ) in the inepenent Rayleigh faing channel case. We now consier two numerical examples to illustrate how cooperation influences the iniviual power requirements of each source as well as the total power requirement for both sources in the faing channel scenario. We first consier the case where all of the channels are moele as inepenent Rayleigh faing an where E[g 1 ] = E[g ] an E[h 1 ] = E[h 1 ], i.e., statistically symmetric channels. Figure shows a particular example of this case where, for a fixe choice of cooperation ratios, each source s transmit powers are calculate for each set of channel realizations an these powers are average over 1 iterations. Figure shows that there is an optimal level of mutual cooperation with fixe α 1 > an α > such that the average total require transmit power is minimize. Figure 5 shows that similar results are obtaine in the asymmetric faing case with the primary ifference being that contours are skewe such that minimum total power operating point requires the source with the statistically user cooperation ratio (B) User 1 Power (P user cooperation ratio (B) User Power (P user cooperation ratio (B) Total Power (P tot Fig.. Two-source cooperative transmission power requirements for the case where g 1 /σ w = g /σ w =1B an h 1 /σ w = h 1 /σ =B. Note that the minimum total transmit power is achieve with no cooperation in this case an there is no choice of {α 1,α } that results in a simultaneous reuction of transmit power for both sources. user cooperation ratio (B) User 1 Power (P user cooperation ratio (B) User Power (P user cooperation ratio (B) Total Power (P tot Fig. 3. Two-source cooperative transmission power requirements for the case where g 1 /σ w =1B, g /σ w =B an h 1 /σ w = h 1 /σ =B. Note that the minimum total transmit power is achieve when source cooperates (α.9) an source 1 oes not cooperate (α 1 =). Also note that an there is no choice of {α 1,α } that results in a simultaneous reuction of transmit power for both sources.

5 stronger channel (source, in this case) to have a higher cooperation ratio than the source with the weaker channel. It is beneficial, in terms of average total an iniviual require powers, for both sources to cooperate even in the asymmetric faing case. V. CONCLUSIONS This paper consiers the problem of efficient power allocation in a wireless communication system with two cooperating sources communicating inepenent information over orthogonal subchannels to one estination. We evelope a framework for power allocation in this scenario aroun the concept of cooperation ratios an erive expressions for the transmit power require by each source to achieve their SNR targets as a function of these cooperation ratios. For the system moel an protocol escribe in Section II, we show that cooperation can reuce the total require transmit power but a simultaneous reuction of the require iniviual transmit powers P 1 an P with respect to the noncooperative case is not possible for any choice of cooperation ratios. This implies that only the source with the stronger channel shoul cooperate when the channels are time-invariant. When the channels are inepenently faing, our results show that cooperation can benefit both sources in terms of reucing their average require iniviual transmit powers E[P 1 ] an E[P ] as well as the average total power even when the sources face statistically symmetric channels to the estination. In light of the time-invariant channel results, these results imply that, even though it is instantaneously suboptimal for both sources to cooperate with fixe non-zero cooperation ratios, mutual cooperation is beneficial on average for both sources when the channels are inepenently faing. REFERENCES [1] A. Senonaris, E. Erkip, an B. Aazhang, Increasing uplink capacity via user cooperation iversity, in Proceeings of the 199 IEEE International Symposium on Information Theory, (Cambrige, MA), p. 15, August [] J. Laneman an G. Wornell, Distribute space time-coe protocols for exploiting cooperative iversity in wireless networks, IEEE Transactions on Information Theory, vol. 9, pp. 15 5, October 3. [3] A. Stefanov an E. Erkip, On the performance analysis of cooperative space time coe systems, in Proceeings of the IEEE Wireless Communications an Networking Conference (WCNC), vol., pp , 3. [] K. Azarian, H. El Gamal, an P. Schniter, On the achievable iversity-vsmultiplexing traeoff in cooperative channels, in Proceeings of the 3th Annual IEEE Conference on Information Sciences an Systems (CISS), (Princeton, NJ), pp. 95 9, March [5] T. Hunter, S. Sanayei, an A. Nostratinia, The outage behavior of coe cooperation, in Proceeings of the IEEE International Symposium on Information Theory, (Chicago, IL), p. 7, June 7-July. [] I. Maric an R. Yates, Cooperative multihop broacast for wireless networks, IEEE Journal on Selecte Areas in Communications, vol., August accepte to appear in IEEE Journal on Selecte Areas in Communications. [7] J. Laneman, G. Wornell, an D. Tse, An efficient protocol for realizing cooperative iversity in wireless networks, in Proceeings of the IEEE International Symposium on Information Theory (ISIT), (Washington, DC), p. 9, June user cooperation ratio (B) Average User 1 Power (P user cooperation ratio (B) Average User Power (P user cooperation ratio (B) Average Total Power (P tot Fig.. Two-source cooperative transmission power requirements for the case where all channels are inepenent an Rayleigh istribute, E[g 1 ]/σ w = E[g ]/σ w =1B an E[h 1 ]/σ v = E[h 1 ]/σ v =B. The minimum average total transmit power is achieve when α 1 = α.7 in this case. user cooperation ratio (B) Average User 1 Power (P user cooperation ratio (B) Average User Power (P user cooperation ratio (B) Average Total Power (P tot Fig. 5. Two-source cooperative transmission power requirements for the case where all channels are inepenent an Rayleigh istribute, E[g 1 ]/σ w =1B, E[g ]/σ w =B, an E[h 1 ]/σ v = E[h 1 ]/σ v =B. The minimum average total transmit power is achieve when α 1.3 an α. in this case.