Optimal Resource Allocation for IR-HARQ

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1 Optimal Resource Allocation for IR-HARQ Chaitanya Tumula V. K. and Erik G. Larsson Linköping University Post Print.B.: When citing this work cite the original article. 00 IEEE. Personal use of this material is permitted. However permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. Chaitanya Tumula V. K. and Erik G. Larsson Optimal Resource Allocation for IR-HARQ 0 Proceedings of the IEEE Swe-CTW. Postprint available at: Linköping University Electronic Press

2 Optimal Resource Allocation for IR-HARQ Tumula V. K. Chaitanya and Erik G. Larsson Dept. of Electrical Engineering ISY Linköping University Linköping Sweden. Abstract In this work we first provide an exact closed-form expression for the packet probability PDP in incremental redundancy IR based hybrid automatic repeat request HARQ schemes with a limit on the maximum number of transmissions. In the later part we extend our work in [5 and consider the optimal resource allocation problem of minimizing the PDP under the constraints of an average transmit power and total number of channel uses. We provide two approaches to solve this optimization problem and compare their performance with the solution given in [5. I. ITRODUCTIO Hybrid automatic repeat request HARQ schemes are the retransmission schemes in which conventional ARQ mechanisms are combined with forward error correction FEC schemes to improve the throughput efficiency [. In type-i HARQ schemes the receiver discards the information content of a data packet if it is received in error using error detection methods such as cyclic redundancy check. In type-ii HARQ schemes the receiver combines the information it received across different transmissions about a data packet. Type-II HARQ schemes are mainly classified into: Chase combining CC schemes: In these schemes the same codeword is transmitted to the destination during the retransmissions. The receiver uses maximal-ratiocombining MRC to decode a data packet. Incremental redundancy IR schemes: In these schemes additional parity bits are sent to the destination in each retransmission. The receiver uses code combining to decode a data packet. A. Related Work Coding performance of both CC and IR based HARQ schemes was presented in[. In[3 a cross-layer optimization problemwassolvedforbothcc andirbasedharq schemes with outdated channel state information at the transmitter. A fixed outage probability analysis of HARQ schemes in block-fading channels was presented in [4. However in [4 the authors assumed that the same number of channel uses as well as the same transmission power is used in different transmission rounds. In our previous work [5 and in [6 the optimization problems of minimizing the packet probability PDP and the bit error rate BER under an average transmit power constraint for IR-HARQ schemes were considered respectively. In the works of [ [5 and [6 This work was supported in part by the Swedish Research Council VR the Swedish Foundation for Strategic Research SSF and ELLIIT. E. G. Larsson is a Royal Swedish Academy of Sciences KVA Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation. b b...b I Encoder c c...c C Interleaver Modulator P P P 3 PL 3 L h h h 3 h L One ARQ round channel uses s s...s Figure. System model for the IR-HARQ scheme considered in this work. the assumption of the same number of channel uses per transmission round was used. B. Contributions In this work we generalize the results of [5 to the case whenthenumberofchannelusesmaydifferfordifferentarq rounds. We specifically provide an exact closed-form expression for the PDP of IR-HARQ schemeswhen the total number of transmissions is limited. The closed-form expression for the PDP is in terms of the upper incomplete Fox s H-function [8. Since these special functions are not mathematically tractable we provide an approximate PDP expression for the case when only two transmissions are allowed. We also extend our previous work in [5 and formulate an optimization problem of minimizing PDP under an average power constraint as well as a limit on the total number of channel uses. We provide two approaches to solve this optimization problem and compare their performance with the solution provided in [5 in which the optimization problem was solved under the average transmit power constraint only. II. SYSTEM MODEL The system model considered for the IR-HARQ scheme is shown in Fig.. I information bits b b...b I are encoded to obtain a code-block with coded bits c c...c C where C denotes the number of coded bits. These C coded bits are then mapped onto a constellation S to obtain modulation symbols s s...s. The modulation symbols are then transmitted to the destination in an incremental redundancy fashion. We assume that the modulation symbols have unit average energy i.e. E s n =. The number of transmission rounds for the IR-HARQ scheme is limited to L and the lth transmission round consists of l channel uses. We also assume that one modulation symbol

3 is transmitted per channel use in each ARQ round i.e. we have L l= l =. We consider a block-fading Rayleigh channel in which the channel gain remains constant in each ARQ round and varies independently between different ARQ rounds. Let h l denote the channel gain for the lth ARQ round and let λ l = E h l l =...L. We assume that the transmitter has statistical knowledge of the channel gains i.e. it has the knowledge about the distribution of channel gains. This assumption is practical as it does not require instantaneous feedback from the receiver. We also assume that the same transmission power is used in all the channel uses of the lth ARQ round and it is equal to P l. With the above described system model we can write the received signal at the destination during the ith channel use of the lth ARQ round as: y li = P l h l s li +e li l =...L i =... l where s li S and e li denote the modulated symbol and the additive white Gaussian noise sample in the ith channel use of the lth ARQ round respectively. We assume that e li l =...L and i =... l are i.i.d. with distribution C 0. Assuming Gaussian signaling and that is large we can write the total accumulated mutual information at the destination till the lth ARQ round as : l I l = m log +P m h m = l m log+p m α m where α m = h m with α m expλ m m =...l. owweprovideaclosedformexpressionforthepdpofthe IR-HARQ scheme with the system model described above. We assume that the feedback signaling from the receiver is based on the total accumulated mutual information at the destination. The receiver sends an acknowledgementack signal after the lth ARQ roundif I l > I andanegativeack ACK signal otherwise. We also assume that the ACK/ACK feedback signaling from the receiver to the transmitter is instantaneous and error-free. So the probability that the packet is in outage after l ARQ rounds can be written as [7: p outl PrI l < I... I < I = PrI l < I 3 and the PDP which is the probability that the packet is in outage after L ARQ rounds is given by: p outl = PrI L < I 4 Proposition. p outl for l L can be expressed in closed-form using the upper incomplete Fox s H-function [8 see Appendix A for its mathematical definition as p outl = K l [ H l+0 0 l δ l 5 Alllogarithms used in this work are with baseunless otherwise specified. Analytical umerical Average SR per Channel Use db Figure. Analytical and numerical packet probability comparison. Parameters for the simulation are L = 3 P = P = P 3 = 0 db λ = λ = λ 3 = λ. = bpcuδ = 0.5 bpcu and δ 3 = bpcu. The average SR per channel use is varied by varying λ. where K l exp l δ m I m for m =...l and l. Proof: See Appendix B. With the convention that p out0 = we define the average transmit power per channel use as: L l= P P l l p outl and the average signal-to-noise ratio SR per channel use L l= in db as 0log P l l λ l p outl 0. Fig. shows the analytical and numerical performance comparison of PDP for an IR-HARQ system with L = 3. The parameters used for the performance comparison are P = P = P 3 = 0 db λ = λ = λ 3 = λ = δ = 0.5 and δ 3 = bits per channel use bpcu respectively. The analytical result in Fig. is obtained by evaluating the expression in 5 by inserting the parameter values. The numerical result curve in Fig. is obtained by generating random channel gains and empirically evaluating the probability in 4. We can see from the figure that the analytical expression result matches closely with the numerical result. III. THE SPECIAL CASE OF L = The closed-form PDP expression provided in 5 does not give much direct insight into the variation of p outl as a function of the resource parameters P l and δ l. Since we are interested in the optimization of resources we now provide an approximate PDP expression which allows us to optimize the resources for the special case of two transmissions. Using the approach given in Appendix A of [5 we can write: The expression in 5 can be evaluated in MATHEMATICA using the program given in Appendix B of [8. 6

4 I LP P γ γ µ µ = P P λ λ I given given µ P + given P p out given I given P P λ λ +γ P +γ P +µ given + P given. 0 Exact Value Using 5 Approximate Value Using Average SR per Channel Use db Figure 3. comparison using exact Fox s H-function evaluation of 5 and the approximate simplified expression of 8. Parameters for the simulation are L = P = P = 0 db λ = λ = λ 3 = λ. = 3 bpcu and δ = bpcu. The average SR per channel use is varied by varying λ. p out = exp δ 7 P λ δ δ ln δ + P P λ λ +O P if = 3 δ = δ = δ δ P P λ λ δ P P λ λ +O P if 3 δ δ δ Fig. 3 shows the comparison of the for the special case of L = using the exact expression given in 5 and the expression given in 8 by neglecting O P terms. We can 3 see from the figure that the approximation using 8 is tight for average SR values higher than 6 db. The optimization results in [5 were obtained under the assumption that the same number of channel uses is used in both ARQ rounds. In this work we remove that restriction and consider the case with δ or equivalently. We neglect the O P terms of 8 in the further analysis. 3 A. Optimization Problem For L = we consider the following optimization problem of minimizing the PDP under the constraints of given average transmit powerper channeluse P given and the total number of available channel uses given. I I I given given min P P P P λ λ given P P λ λ subjectto C :0 P 8 C :0 P C 3 :0 given 9 C 4 : P + given P p out given P given The objective function in 9 is obtained from the PDP expression in 8 and using the fact that + = given. Proposition. Let P P denote the optimal solution to 9 then the constraint C 4 of 9 is active at the optimal solution. Proof: We write the Lagrangian function of 9 as shown in 0 on top of the page. Let P P γ γ µ µ denote the optimal values of 0. Considering the Karush- Kuhn-Tucker KKT necessary conditions [9 we require: µ P + given P p out given P given = 0 and µ 0. To prove that C 4 is active we need to prove that µ L P > 0. ow considering P P γ γ µ µ = 0 we have I I I given given + P P λ λ given P P +γ +µ λ λ given I + given given P λ p out P P P γ γ µ µ = 0. Since P given is bounded above by as seen from C 4 we have γ = 0. Use this in and note that µ = 0 if and only if = = given. However by observing that = given cannot be the optimal solution we have that µ > 0. ow using the result that C 4 is active at the optimal solution we have P given P = givenp given P given p out. Using this we can simplify the optimization problem in 9 to the following: min f P P subjectto 0 P and 0 given 3

5 where I f P = P λ λ I given given I given P λ λ given p out. 4 given P given P The optimization problem in 3 is a mixed-integer non-linear programming MILP problem and one can use MILP solvers to find the optimal solution [0. ow we provide two approaches to solve 3. The first approach solves 3 optimally and in the second approach we provide a suboptimal solution. B. Solutions to 3 First Approach: In the first approach we solve the optimization problem in 3 using the following method. Since isanintegervariableforafixedvalueof onecan reduce the two dimensional optimization problem in 3 to a one dimensional sub-problem in the variable P as follows: p out mingp P P given P given P subject to 0 P P UB 5 where P UB is the upper bound of P for a fixed value of and is given by: P UB = given P given. 6 One can use constrained non-linear optimization techniques [9tofindthesolutionto5.OncetheoptimalsolutionP to 5 is obtained the corresponding PDP can be computed as P = K.g P 7 where K is a constant and is given by: K = f P gp 8 After solving the sub-problem in 5 for all feasible values of we find the optimal solution to 3 by selecting the pair P which gives the smallest p P value. The approach is summarized as follows: Approach : Initialize p opt =. for from to given a Solve the sub-problem in 5 to obtain g P. b Compute p current using 7 and 8. c if p current < p opt p opt = pcurrent. P = P. =. else Increment. end if end for 3 Compute P from using P and. Second Approach: The second approach is similar to the first approach except that we provide a sub-optimal closed form solution for the sub-problem in 5. eglecting the constraint that P is bounded above by P UB following a similar approach as in [5 the stationary points of gp are given by the the following equation: p out given P given P = p out given P given P P λ 9 ow using an approximate Taylor series expansion for p out and neglecting the O P terms we have 3 p out δ 0 P λ P λ Substituting 0 in 9 and simplifying we obtain the stationary points of gp in 5 as the rootsof the following cubic equation: ap 3 +bp +cp +d = 0 where a = 6 zλ b = 4P given zλ 4 z λ c = 3P given z λ + z 3 d = z 3 P given and z = δ. The fact that d 0 guarantees that the cubic equation has at least one positive real root. Following similar approach as in [5 we can prove that there is only one positive real root for the cubic equation in. Moreover this positive real root is feasible it satisfies the constraint in 5 if the following condition is satisfied: P UB δ λ When the condition in is satisfied the optimal P for the sub-problem in 5 is given by the real positive root of. On the contrary if is not satisfied then we have P UB < δ λ = δ P UB λ < which implies that p out in 7 is close. In this case we can choose P = PUB 3 which is the value of P that maximizes the denominator of gp in 5. The second approach is summarized below: Approach : Initialize p opt =. for from to given a Solve for P as follows: if is satisfied Obtain P as the positive real root of.

6 else Obtain P using 3. end if b Compute p current c if p current using 5 7 and 8. < p opt p opt = pcurrent. P = P. =. else Increment. end if end for 3 Compute P from using P and. IV. UMERICAL RESULTS In this section we present numerical results comparing the performance of both approaches presented in Section III-B for solving the optimization problem in 9. Fig. 4 shows the comparison for different spectral efficiency values. For comparison purposes we have also plotted the result with only optimal power allocation of [5 obtained by assuming = =. In Fig. 4a with I = 400 bits and = 000 channel uses i.e. using more number of channel uses to transmit a small number of information bits the solution provided by the optimal resource allocation by both Approach and Approach has similar performance as that of optimal power allocation by assuming = = [5. Intuitively this implies that for low values of I in order to minimize the optimal way to allocate channel uses is to use = = this has also been confirmed from the optimal valueobtainedfromboththeproposedapproaches. We canalsoseefromthefigurethatapproachhaspractically the same performance as that of Approach. Fig. 4b shows the result for a higher value of I i.e. using a smaller number of total channel uses to transmit a larger number of information bits. As we can see in this case the optimal resource allocation number of channel uses and power has better performance than that of just using optimal power allocation assuming equal channel uses. At high average SR values the optimal allocation of channel uses for both the approaches is observed as.5. The gain for optimal resource allocation over only optimal power allocation is about 0.6 db. In terms of complexity Approach involves finding a positive real root of a cubic equation compared to exact one dimensional minimization in Approach for all possible valuesof. 3 Howeverthe solutionin [5involvesthe finding a root of a cubic equation only for = given /. V. COCLUSIOS AD DISCUSSIO In general the closed-form expression provided for the PDP of an IR-HARQ scheme with limited number of retransmissions may not be directly useful for solving the resource allocation problem. Approximate expressions to PDP are required 3 This is the worst-case scenario. When channel statistics are changing slowly one can search only over a subset of all possible values by using knowledge about previous solutions. Approach Approach Solution in [ Average SR per Channel Use db 0 6 a With I = 400 and = 000. Approach Approach Solution in [ Average SR per Channel Use db b With I = 4000 and = 000. Figure 4. comparison using both the approaches presented in Section III-B. For comparison purposes we also presented the result from [5 which assumes an equal number of channel uses per transmission and uses optimal power allocation. The parameters for the simulation are P given = 0 db. The average SR is varied by varying λ = λ = λ. to solve these optimization problems. For the special case of two transmissions and low spectral efficiency values it is optimal to allocate equal number of channel uses for both ARQ rounds. For high target spectral efficiency values the equal number of channel uses has a small performance degradation compared to optimal allocation. APPEDIX A ICOMPLETE FOX S H-FUCTIO The generalized upper incomplete Fox s H-function is defined as [8: Hpq xy r Hxy pq r a γ A...a p γ p A p b η B...b q η q B q = π Mpq xy [sr s ds 4 C where xyp and q are integers with 0 x p and 0 y q. a i b i C and γ i η j A i B j R + for i p and

7 j q. M xy pq [s is the Mellin transform of H xy pq r and is defined as: y Mpq xy [s = i= Γ a i γ i sa i x j= Γb j +β j sb j p i=y+ Γa i +γ i sa i q j=x+ Γ b j β j sb j 5 where Γ is the upper incomplete Gamma function. The contourofintegrationc in4ischosensuchthattheintegral corresponds to Mellin-Barnes integral [. APPEDIX B PROOF OF PROPOSITIO From and 3 we have [ l p outl = Pr m log+p m α m I = Pr = Pr [ [ log log l l +P m α m X m 6 [ l = Pr log Z m = Pr[Y = F Y where X m for m l is a shifted exponential random variable with p Xm x m = exp xm x m. F Y y = Pr[Y y is cumulative distribution function CDF of Y. Using Mellin transform property from [ we can write the Mellin transform of W = l Z m as: M W {p W w}s = l M Zm {p Zm z m }s 7 and the Mellin transform of Z m m l is defined as: ˆ s xm M Zm {p Zm z m }s = p Xm x m dx m 0 exp = P m λ ˆ m s xm exp x m P m λ m a = exp s Γ s δ m + dx m 8 in a we have used the result from [ pp The region of convergence for the result in 8 is R{s} > 0. ow using 7 and 8 we have M W {p W w}s = K l [ s l s Γ + δ m 9 where K l and are defined in Proposition. The probability density function PDF of W = l Z m is given by the Mellin-Barnes integral as: p W w = π = K l π C:R{s}>0 C:R{s}>0 s Γ + δ m = K l [ w H l0 0l M W {p W w}sw s ds l w s ds δ l δ l We can write the PDF of Y = logw from [8 A.0 as : [ p Y y = K l.ln.h l0 y 0l... δ l 3 and the CDF of Y can be expressed from [8 A.4 as: F Y y = K l [ H l+0 y l δ l from 6 and 3 we can obtain 5. REFERECES 3 [ S. Lin D. J. Costello Jr. and M. J. Miller Automatic-repeat-request error control schemes IEEE Commun. Mag. vol. pp. 5-6 Dec [ J.-F. Cheng Coding performance of hybrid ARQ schemes IEEE Trans. Commun. vol. 54 pp June 006. [3 W. Rui and V. K.. Lau Combined cross-layer design and HARQ for multiuser systems with outdated channel state information at transmitter CSIT in slow fading channels IEEE Trans. Wireless Commun. vol. 7. no. 7 July 008. [4 P. Wu and. Jindal Performance of hybrid-arq in block-fading channels: a fixed outage probability analysis IEEE Trans. Commun. vol. 58 no. 4 pp. 9-4 Apr. 00. [5 T. V. K. Chaitanya and E. G. Larsson Outage-optimal power allocation for hybrid ARQ with incremental redundancy IEEE Trans. Wireless Commun. vol. 0 no. 7 pp July 0. [6 H. Liu L. Razoumov and. Mandayam Optimal power allocation for type II H-ARQ via geometric programming Conference on Information Sciences and Systems CISS Mar [7 S. Sesia S. G. Caire and G. Vivier Incremental redundancy hybrid ARQ schemes based on low-density parity-check codes" IEEE Trans. Commun. vol. 5 no. 8 pp. 3-3 Aug [8 F. Yilmaz and M.-S. Alouini Product of shifted exponential variates and outage capacity of multicarrier systems in Proc. European Wireless Conference May 009. [9 J. ocedal and S. J. Wright umerical Optimization nd ed. Springer 006. [0 M. R. Bussieck and S. Vigerske MILP solver software available online at [ B. D. Carter and M. D. Springer The distribution of products quotients and powers of independent H-function variates SIAM Journal on Applied Mathematics vol. 33 no. 4 pp Dec [ I. S. Gradshteyn and I. M. Ryzhyk Table of Integrals Series and Products 7th ed. Academic Press 007.

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