FOR energy limited data networks, e.g., sensor networks,

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1 578 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., DECEMBER 009 An Optima Power Aocation Scheme for the STC Hybrid ARQ over Energy Limited Networks Hongbo Liu, Member, IEEE, Leonid Razoumov, Member, IEEE, Narayan Mandayam, Feow, IEEE, and Predrag Spasojević, Member, IEEE Abstract In this paper, we show that for STC (Space-Time Coded) Hybrid-ARQ (Automatic Repeat request) schemes with (re)transmission power contro over independent Rayeigh bock fading channes, the probem of optimizing energy efficiency with a PER (Packet Error Rate) constraint can be soved as a geometric programming probem. The optimum transmit power increases super-ineary with each requested retransmission and the fraction of the average power optimay aocated to each ARQ round ony depends on the incrementa diversity gain. The energy savings increases with a decrease in the PER targets and decreases with an increase in the diversity gain. Index Terms Energy imited networks, geometric programming, hybrid ARQ, space-time codes. I. INTRODUCTION FOR energy imited data networks, e.g., sensor networks, one of the goas is to deiver as much information as possibe with a imited energy suppy and a maximum deay constraint. For data transmissions with oose deay constraints, the ARQ (Automatic Repeat request) scheme is often used in addition to an FEC (Forward Error Correction) scheme, which is known as Hybrid ARQ, to provide adaptive data rates, and hence improve the energy efficiency. In the foowing, we refer to the uncoded information message as a source packet" and the successive transmissions of the coded bocks of the same source packet as ARQ rounds". The traditiona Hybrid ARQ scheme is a constant power transmission scheme. A few power adaptation schemes have been proposed to increase the throughput [], reduce energy consumption [], or provide extra diversity gain [3] by making the transmit power of the -th ARQ round a function of. In [3], for ong-term static MIMO (Mutipe Input Mutipe Output) channes, i.e., the channe coefficients remain constant during a ARQ rounds and change to new independent vaues with each new source packet, an asymptoticay optima power contro agorithm is proposed that yieds very significant diversity advantage. However, for short-term static channes, where the channe coefficients remain constant during each Manuscript received March 8, 008; revised October 4, 008 and March 6, 009; accepted June, 009. The associate editor coordinating the review of this etter and approving it for pubication was M. Torak. H. Liu, N. Mandayam, and P. Spasojević are with WINLAB, Department of Eectrica and Computer Engineering, Rutgers University, North Brunswick, NJ 0890 USA (e-mai: {hongbo, narayan, spasojev@winab.rutgers.edu). L. Razoumov is with AT&T Labs, 80 Park Ave, Forham Park, NJ 0793, USA (e-mai: eor@research.att.com). The materia in this paper was presented in part at Conference on Information Sciences and Systems (CISS) 004 and CISS 005. This work was supported in part by the NSF under Grant No. FMF Digita Object Identifier 0.09/TWC /09$5.00 c 009 IEEE ARQ round and change independenty for the next ARQ round, no optima power adaptation scheme is proposed. Different from previous works, we propose a power adaptation agorithm where the power eve at each ARQ round is optimized for the short-term static channe variation based on the ong term channe statistics. Our goa is to choose the transmit power for the -th ARQ round to minimize the average energy consumption under a PER (Packet Error Rate) constraint with the maxima number of ARQ rounds to be fixed to L over an L-bock Rayeigh fading STC (Space-Time Coded) channe. We show that our probem can be formuated and soved as a GP (Geometric Programming) probem and a cosed-form soution can be obtained. We aso show that the optimized transmit power increases in a super-inear manner each time a retransmission is requested and the fraction of the average energy optimay aocated to the -th ARQ round ony depends on the incrementa diversity gain. This paper is organized as foows: In Section II, we provide the channe mode and some preiminaries used in this work. In Section III, we show how the optimization probem can be formuated and soved as a GP probem. In Section IV, simuation vaidations for the optima power aocation strategy are presented and we summarize our work in Section V. II. CHANNEL MODEL AND THE PRELIMINARIES A. Channe Mode for Hybrid ARQ There are severa variations of the Hybrid ARQ schemes, i.e., type I [4], type II with IR (Incrementa Redundancy) [5] and Chase combining [6]. In this paper, an error-free and zerodeay ARQ feedback channe is assumed for a ARQ schemes. If a decoding error occurs, a one-bit retransmission request (NACK) is sent back to the transmitter. We consider a frequency fat fading channe with no CSI (channe state information) at the transmitter and perfect CSI at the receiver. For a STC channe with M T transmit antennas and M R receive antennas, the channe state matrix H C MR MT,,,Lremains constant during the -th ARQ round and changes independenty for different ARQ rounds. Each ARQ round takes T channe uses. Define x,t C MT, y,t C MR,andw,t C MR, =,,L,t =,,T as the transmitted signa, received signa, and corresponding channe noise vector, respectivey. The channe noise is assumed to be temporay and spatiay white with i.i.d. compex Gaussian entries CN(0,N 0 ).LetE be the transmit power per channe use, which remains constant for the -th ARQ

2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., DECEMBER round, and the energy consumption for the -th ARQ round is E T. The channe mode is defined as E y,t = H x,t + w,t, () M T which corresponds to a short term static channe in [3]. It is aso referred to as the L-bock fading Hybrid-ARQ mode in this paper. A power aocation scheme for an L-bock fading Hybrid- ARQ is defined as E =(E,E,E L )=(ρu,ρu,,ρu L )=ρu, where ρ = E and u = E E,and represents the norm of the vector. B. Average Power In this paper, the average power is defined as the average energy normaized by T.Define the probabiity of the event that a NACK is received at the -th round as P,andetP 0 = by definition. Let E[ ] denote the expectation of a random variabe. Foowing [3], the average power per source packet is [ T ] Ē = E E = E P, () where T is a random number of retransmissions for each source packet. C. Diversity Gain The optima diversity-mutipexing tradeoff for ARQ MIMO channes has been proposed in [3]. In this paper, we use a different approach to obtain the diversity gain. For an Lbock fading Hybrid ARQ, we define the term incrementa diversity gain. For a given power aocation scheme E = ρu, the diversity gain at the -th ARQ round is defined as d = im og P (ρu), (3) og ρ where P (ρu) is the accumuative PER after the -th ARQ round. The incrementa diversity gain, which represents the diversity gain corresponding to the -th ARQ round, is defined here as D = d d = im where d 0 =0by definition. og P (ρu) og P (ρu), og ρ D. Probabiity of Errors in Hybrid ARQ The average energy consumption is cosey reated to the PER. For each ARQ round, the CSI is unknown to the sender, and therefore, the resuts for the open-oop PER anaysis can be appied. We can derive the PER for both type I and type II L-bock fading Hybrid-ARQ mode based on the anaysis for STC in [7]. Let E denote the event that a decoding error occurs at the th ARQ round. We assume that the undetected decoding error can be ignored. Given a sequence of channe state matrices Fig.. Ω ( a, a ) σ σ Error event decision regions. x Ω ( a σ, a ) σ H, H,, H L, the probabiity of sending a NACK at the -th ARQ round for =,,L is given in equation (4). P (ρu H, H, H )=Pr(E H )Pr(E E, H, H ) Pr(E E, E,, E, H, H, H ) = Pr(E,, E H, H, H ). (4) We use superscripts I and II to denote the type I and the type II Hybrid-ARQ respectivey. For type I Hybrid ARQ, ony the atest received packet is used for decoding. For Chase combining and type II Hybrid ARQ, the code bocks from the ARQ round through are a combined and decoded as one codeword using a maximum ikeihood (ML) decoder. The Chase combining Hybrid ARQ is considered a specia case of the type II Hybrid-ARQ and, therefore, is not discussed separatey. Proposition : The error probabiity of an L-bock fading type II Hybrid ARQ after the -th ARQ round has the same diversity order as the error probabiity of decoding a codeword with fading bocks at the -th ARQ round aone without considering decoding errors at previous ARQ rounds, i.e., og Pr(E,, E ) d = im = im og ρ x og Pr(E ). (5) og ρ Proof: For an L-bock fading type II Hybrid ARQ with maximum ikeihood (ML) decoding rue, the error probabiity can be bounded as Pr(E H, H, H ) Pr(E,, E H, H, H ) Pr(E H, H, H ). (6) The upper bound is obvious because the probabiity of the joint event is equa to or ess than the probabiity of an individua event. The ower bound can be obtained based on the properties of the Eucidean distance based ML decoding as showninfig.,wherex i = x i,a i = s i,i =,,,and x i is the received signa, s i is a super symbo composed of a different symbos between two codewords, Ω is the error decision region, and σ i is the noise power at the i-th ARQ

3 570 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., DECEMBER 009 round. We have Pr(E, E H, H )= p(x,x )dx dx Ω Ω p(x,x )dx dx Ω = p(y )dy = Ω Pr(E H, H ), x with change of variabes y = + x σ. Equation (7) is σ based on the property of muti-dimensiona Gaussian distribution. With the same anaysis, it foows that Pr(E, E,, E H, H, H )= p(x)dx Ω p(x)dx = (8) Ω Pr(E H, H, H ). Taking the expectation over a channe states for equation (6), equation (5) then foows. Proposition : The maximum diversity order of an Lbock fading Hybrid-ARQ channe is equa to M T M R L and the maximum incrementa diversity is D = D = = D L = M T M R. Proof: For the type I Hybrid ARQ, a error events E are mutuay independent, therefore, the PER at the -th ARQ round is I (E) =E[P (E H, H, H )] = E[Pr(E k H k )] = Pr(E k )= P I Pr(E k ) where Pr(E k ) is the PER for the k-th ARQ round. By appying [7, equation (3)], the asymptotic PER bound of the type I Hybrid-ARQ after the -th ARQ round is P I (E) 4 [ Ek 4N 0 ] ηmin M R e E min [ ηmin i= (7) (9) ] MR λ i, (0) where e is the error sequence, E min is the set of error sequences with minimum diversity, λ i is the eigenvaues derived from the STC channe matrix [7, equation (3)], and η min is the minimum diversity of a singe bock fading channe. With optimum code design, maximum η min = M T is achieved, and Proposition foows. For the type II Hybrid-ARQ, based on Proposition, to obtain the maximum diversity order and the incrementa diversity gain, we ony need to consider the decoding error corresponding to an bock fading channe. To obtain the PER with a power aocation scheme E = (ρu,ρu,,ρu L ) over an bock fading channe, we consider an equivaent transmission scheme, in which the transmit power is ρ for each transmission whie the fading gain is scaed by u k for the k-th fading bock. The asymptotic PER bound of the type II Hybrid-ARQ after the -th ARQ round is [7, equation (3)] P II (E) [ ] ηmin M ρ R 4 4N 0 e E min [ η k i= ] MR λ (k) i u k, () where η min = min e η k and η k is the rank of the transformed STC channe matrix defined in [7, equation (0)] for the k-th transmission. The maximum diversity gain that can be achieved is max{η min = LM T when η k = M T. Substituting η k with M T eads to Proposition. We note that the maximum diversity gains for the type I and the type II Hybrid-ARQ are same. The packet combining does not increase the diversity order. From equations (0) and (), the unified approximation of the PER for STC Hybrid ARQ schemes with maximum diversity order at high SNR (Signa-to-Noise Ratio) can be written as ( ) Dk Ek P (E,,E ) A, () where the bound for A for the type I and the type II Hybrid- ARQ can be obtained from equations (0) and (). In the remainder of this paper, we wi use the above expression for the transmission probabiity in deriving optima power aocation strategies for the Hybrid ARQ system. N 0 III. OPTIMAL POWER ALLOCATION The optima power aocation probem that minimizes the average power under a PER constraint can be formuated as { min E = E P (E,,E ) E,E,,E L subject to P L (E,,E L ) P max e where P 0 =by definition.,e > 0,,,L, (3) The above optimization probem can be written as a GP probem [8] [0] in the standard prima form after appying approximation (). min { f 0 (x) =x + A x x D x + A x 3 x D x D + subject to A L L + A L x L L i= x D x D Pe max,x > 0,,,L, (4) where x =(x x x L ),x = E /N 0, =,,L and D is the incrementa diversity gain for the -th ARQ round. This is a GP with a zero degree of difficuty and thus has a unique soution. Because of its specia form, we can obtain the cosed-form soution by soving its dua program, which is formuated as L ( ) δ ( ) δl+ A AL max { v(δ) = δ subject to δ δ =0,δ P max e L+ j=+ δ j D =0,,,L, (5) where δ =(δ,,δ L+ ) is the design variabe for the dua

4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., DECEMBER program, which can be soved from the inear constraints as δ = D j=+ ( + D j) j= ( + D, =,, L, j) δ L = δ L+ = D L j= ( + D j), j= ( + D j). (6) From the prima dua reation, the prima program and its optima soution x =(x,x,,x L ) satisfy L ( ) δ ( ) f 0 (x )=v(δ A δ L+ A L )=, (7) δ Pe max where δ =(δ,δ,,δ L+ ). Let u (x) =A x j= x Dj j,thenf 0 (x) =u (x) + u (x) + + u L (x), andδ = u (x)/f 0 (x) is defined as the proportion of the average energy consumption for the -th ARQ round reative to the tota energy consumption. According to the duaity theory [8, Theorem (iv)], we have u (x )=v(δ )δ = f 0 (x )δ, =,,,L. (8) And the optima soution satisfies x = f 0(x )δ, x = A D A D ( + D ) (x )(+D ),=,,L. (9) Note that the PER approximation does not hod for SNR < and, therefore, the optimization scheme cannot be appied with extremey ow SNRs. Aso note that the optimization scheme does not change the diversity gain, because the incrementa diversity gain for each ARQ round is a constant. The theoretica bound for the A is not tight enough to derive the accurate vaue of the A. Therefore, it is obtained from the PER statistics of the channe measurement or simuations in practice. A. Impact of the diversity gain To simpify the notation, we assume equa incrementa diversity gains D = D, =,,L and either the number of ARQ rounds L or D is arge. For conventiona Hybrid ARQ with equa power aocation for each ARQ round, i.e., x = x = = x L = ρ and a target PER P e, based on equation () and (), the average power Eav NPC is cacuated as = ρ A ρ D( ), (0) E NPC av ( ) DL P where the transmit power ρ = e A L, which comes from the PER constraint A L i= x D P e and A 0 = by definition. We use superscript NPC to represent no power contro case. For the optima power aocation scheme and a arge D, we can approximate the equation (6) as δ D, =,,,L. () ( + D) Frame Error Rate Equa Power Aocation Optima Power Aocation Average Power (db) Fig.. PER for equa power aocation scheme and optimum power aocation scheme for type I Hybrid ARQ. With the equa power aocation scheme and a arge D, δ NPC A ρ D( ), =,,,L. () For both optima and equa power aocation schemes, with arger incrementa diversity gain, i.e., arger number of transmitter and receiver antennas, more energy is aocated to the first ARQ round, i.e., δ and δnpc as D. Hence the Hybrid-ARQ scheme converges to a singe transmission scheme with an increase of the diversity gain. This fact suggests that from the energy saving point of view, a singe transmission scheme is good enough for a system with arge diversity gain. Compared with the δ, δnpc converges faster as a function of D. Hence, as the incrementa diversity gain increases, the conventiona Hybrid ARQ scheme converges faster to a singe transmission scheme than the optima scheme in terms of the average energy consumption, and both Hybrid ARQ schemes do not provide much energy savings over singe transmission schemes. Therefore, the performance gain of an optima power aocation scheme over an equa power aocation scheme diminishes as incrementa diversity gain increases. IV. APPLICATIONS In the foowing, we appy the optima power aocation scheme in simuated Hybrid ARQ systems. To iustrate the performance gain against the no power contro Hybrid-ARQ systems, we appy the type I Hybrid-ARQ and Chase combining Hybrid-ARQ over a x SIMO (Singe Input Mutipe Output) Rayeigh bock fading channe with an 8-states 4-PSK STTC coding scheme defined in [, Fig. 5]. The diversity and coding gain parameters of the PER, i.e., A,D, =, are obtained from the PER statistics based on simuations. We obtain PER curves for 5 cases, where the power difference between the two ARQ rounds is -6 db, -3 db, 0 db, 3 db and 6 db, and find the best fit for A,D,A,D. Next, we cacuate the vaue of x,x of a set of PER targets Pe max = {0., 0.05, 0.0, 0.005, 0.00, , For each set of x,x, we obtain the PER vaue by simuation

5 57 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., DECEMBER 009 Frame Error Rate Equa Power Aocation Optima Power Aocation Average Power (db) Fig. 3. PER for equa power aocation scheme and optimum power aocation scheme for Chase combining Hybrid ARQ. to evauate the power savings. The resuts are shown in Figure and 3. Note that, for extremey high PER ike 0.5, the optimization scheme is not appicabe. In Fig. and Fig. 3, the resuts show that at a target PER of 0 4, the optima power aocation scheme can provide a gain of up to 3 db for the type I Hybrid-ARQ and.5 db for the Chase combining Hybrid-ARQ. V. CONCLUSION In this paper, we have provided a method to minimize the average energy consumption for Hybrid ARQ systems over an L-bock Rayeigh fading channe whie maintaining a target PER. The optimization probem was formuated and soved as a geometric programming probem. The Lagrangian dua formuation was used to derive cosed-form expressions for the optima objective function and the optima power aocation. Compared with a traditiona Hybrid-ARQ scheme with equa power aocation for a ARQ rounds, the optima scheme achieved significant power savings for channes with ow diversity gains. The simuation resuts showed, at a target PER of 0 4, a gain of up to 3 db for type I Hybrid-ARQ and.5 db for Chase combining Hybrid-ARQ were achieved over an STTC (Space-Time Treis Code) coded SIMO channe with a maximum of two (re)transmissions. The anaytica resuts showed that a arger gain was achieved for a ower PER target, and the energy savings obtained from the optimization scheme decreased with a higher incrementa diversity gain. REFERENCES [] S.-H. Hwang, B. Kim, and Y.-S. Kim, A Hybrid ARQ scheme with power ramping," in Proc. Veh. Techno. Conf. (VTC 00), vo. 3, Oct. 00, pp [] Z. Sun and X. Jia, Energy efficient Hybrid ARQ scheme under error constraints," Wireess Persona Commun., vo. 5, pp , 003. [3] H. E. Gama, G. Caire, and M. O. Damen, The MIMO ARQ channe: diversity-mutipexing-deay tradeoff," IEEE Trans. Inf. Theory, vo. 5, pp , Aug [4] S. Lin, D. J. C. Jr., and M. J. Mier, Automatic-repeat-request error contro schemes," IEEE Commun. Mag., vo., no., pp. 5-7, Dec [5] D. M. Mandebaum, An adaptive-feedback coding scheme using incrementa redundancy," IEEE Trans. Inf. Theory, vo. 0, pp , May 974. [6] D. Chase, Code combining a maximum-ikeihood decoding approach for combining an arbitrary number of noisy packets," IEEE Trans. Commun., vo. 33, pp , May 985. [7] M. Chiani, A. Conti, and V. Trai, A pragmatic approach to spacetime coding," in Proc. IEEE Internationa Conf. Commun. 00 (ICC), vo. 9, June 00, pp [8] R. J. Duffin, E. L. Peterson, and C. Zener, Geometric Programming: Theory and Appication. John Wiey & Sons, Inc., 967. [9] S. Rao, Optimization Theory and Appications. John Wiey & Sons, Inc., 983. [0] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 004. [] V. Tarokh, N. Seshadri, and A. R. Caderbank, Space-time codes for high data rate wireess communication: performance criterion," IEEE Trans. Inf. Theory, vo. 44, pp , Mar. 998.