Physical layer reliability vs ARQ in MIMO block-fading channels

Transcription

1 Physical layer reliability vs AQ in MIMO block-fading channels Marie Zwingelstein-Colin and Mérouane Debbah University Lille Nord de France IEMN, UM 8, F-9 Valenciennes, France SUPELEC Alcatel-Lucent Chair on Flexible adio rue Joliot-Curie 99 IF SU YVETTE CEDEX France Abstract In today s wireless communication systems, automatic repeat request AQ is implemented at the MAC layer in order to retransmit packets that have been erroneously transmitted at the physical PHY layer. Following a joint PHY- MAC design, information provided by the AQ scheme can be exploited at the PHY layer in order to improve the system s performance. This paper extends the work presented in [] in the context of channels to the context of MIMO blockfading channels. Based on statistical channel knowledge at the transmitter, it provides an analysis of the natural tradeoff that exists between the PHY layer transmission rate and the number of AQ retransmissions. We derive a very accurate analytical formulation of the optimum transmission rate and, equivalently, the optimum PHY packet error-rate that maximizes the goodput, as a function of the system parameters, namely the SN, the number of antennas and the diversity order of the channel. Interestingly, we find that the PHY layer has to be made more reliable for MIMO channels than for channels, and also that the MIMO AQ system is less sensitive to a wrong choice of the rate of transmission than the AQ system. I. BACOUND AND MOTIVATION Since several years, cross-layer design have drawn the research community s attention to enhance wireless systems performance. Cross-layer design can follow two approaches: firstly, the sharing of the PHY and MAC layers knowledge with higher levels, also known as cross-layer networking []; secondly,the use of the MAC layer to optimizethe PHY layer. This paper concentrates on the second approach, in the context of MIMO block-fading channels whith AQ retransmission. AQ is a well known feeddback-based technique that consists in retransmitting packets in case of unsuccesfull transmissions, which is particularly advantageous when the instantaneous channel quality is unknown to the transmitter[]. Within AQ, a natural tradeoff exists between the PHY layer reliability and the AQ retransmission rate: on the one hand, if the PHY layer tries to transmit at a high rate, a lot of packets will be detected as erroneous and a lot of retransmissions will be necessary, which degrades the rate at which information is successfully transmitted - also known as the goodput. On the other hand, if the PHY layer is made very reliable, almost no retransmission will be necessary, but the rate of transmission ensuring such high reliability will be quite low. [] and [] have analyzed this tradeoff in the context of blockfading channels. In particular, they have studied how the PHY transmission rate should be adjusted in order for the goodput tobe maximized.theworkpresentedin thispapercanbeseen as a generalization of [] to the MIMO case. Previous work on the MIMO AQ channel has majoritarily focused on code design: the fundamental tradeoff between diversity, throughput and maximum number of retransmissions was derived in [] after its derivation is the case in [] for discrete input constellations. Extensions to multi-bit feedback and imperfect feedback have been derived in [7] and [8] respectively. In this paper, we focus on the performance analysis of a MIMO AQ block-fading channel that, based on statistical channel knowledge, initiates rate adaptation to maximize the goodput. In particular, we derive a very accurate analytical formulation of the optimum PHY transmission rate and, equivalently, the optimum PHY packet error-rate that maximizes the goodput, as a function of the system parameters, namely the SN, the number of antennas and the diversity order of the channel. Interestingly, we find that the PHY layer has to be made more reliable for MIMO channels than for channels, and also thatthemimoaq systemisless sensitivetoawrongchoice of the rate of transmission than the AQ system. II. CHANNEL MODEL We consider a MIMO ayleigh fading channel, with n a antennasat the transmitterandn a antennasat the receiver.the channel is assumed to change independently from one block of data symbols to the other, and to remain constant within each block block-fading scenario. One codeword packet is supposed to span successive blocks, where is representative of the time or frequency selectivity of the channel if time selectivity is considered, the value = corresponds to a slow-fading scenario while higher values of englobe the fast-fading case. If frequency selectivity is considered, = corresponds to a flat channel, whereas higher values reflect a dispersive channel. The channel is assumed to be perfectly known at the receiver, but only statistically known at the transmitter, which is a realistic assumption, especially in a fastfading and/or dispersive environment. We consider a simple AQ scheme with perfect error detection at the receiver: after processing a received codeword, and in case successfull decoding is detected, a one-bit AC signal is sent back to the receiver for aknowledgement, whereas a one-bit NA is

2 fed back if unsuccessfull decoding is detected. In the latter, the transmitter retransmits the packet, until positive AC. No attempt is made by the receiver to correct errors based on potentially previous versions of erroneously received packet, and the cost of the AC/NA feedback will be neglected. At block number k, the l-th received data symbol can be written: SN y l = H k x l +w l n a where x l C na is the l-th transmitted data symbol E{x l x H l } =, w l CN,I na represents the additive aussian noise and H k C na na is the channel matrix at block number k whose independent elements CN,. We are interested in analyzing the tradeoff between the transmission rate and the packet error rate, when one wants to optimize the goodput. For this purpose, we make the assumption that a strong channel code is used, which leads to the source of error for the transmission of a packet being limited only to the outage of the mutual information. In that case, the packet error rate ǫ is simply ǫ = Pr[I,n a, SN ] where is the transmission rate in bit/symbol and I,n a, SN = I+ k= log SN n a H k H H k is the channel mutual information. Note that no attempt is made to optimize the input covariance matrix. Only rate adaptation is performed based on channel s statistics. We denote X the random variable which represents the number of transmission attempts of each packet. In accordance with the block-fading scenario, successive blocks -and thus packets - errors are independent, so the goodput is simply the ratio of over the average number of transmission attempts E{X}: = E{X} Under error-free acknowledgment, the probability of i packet transmission is simply Pr[X = i] = ǫ i ǫ, that is X is a geometrically distributed random variable of mean ǫ. Consequently, the goodput can be expressed = ǫ where the dependency of with ǫ follows equation. III. AUSSIAN APPOXIMATION The analysis of equation requires first to examine the dependency of with ǫ, and also to investigate how this dependency evolves with the system parameters n a, and SN. For this purpose, we propose to approximate the mutual information I,n a, SN by a aussian random variable with the same mean and variance, noted µ,n a, SN and σ,n a, SN respectively. Including this aussian Obviously we have µ,n a, SN = µ,n a, SN and σ,n a, SN = σ,n a,sn. a.. Fig. : β vs SN and b MIMO approximation to the packet error rate equation we get Q ǫ Q µ,na,sn σ,n a,sn µ,na,sn σ,n a,sn µ µ obviously, ǫ = when = µ. As the number of fading coefficients increases with n a and, the central limit theorem tells us that the aussian approximation will be accurate for high n a and. But how high? To answer this question, we quantify the accuracy of the approximation by comparing the values of two caracteristic statistical parameters, namely and β,n a, SN def = [µ,n a, SN] [µ,n a, SN] β,n a, SN def = µ,n a, SN [µ,n a, SN] where µ i denotes the central moment of order i of the distribution, for the actual and for the aussian distributions it can be shown that β = and β = for a aussian distribution, independently of the values of its central moments. Figures and representsthe valuesof β and β for {,,,}, as a function of the SN and n a {,}. The closer β and β are to and respectively, the more accurate the aussian approximation is. In the case, β and β are not very close to and respectively, especially for {,} and SN values < db or > db. The aussian approximation is thus not very accurate, and it can only be used as an indicator in the case. Now looking at the MIMO case, we see that the accuracy of the approximation is greatly improved, even if the selectivity of the channel is reduced to =. The approximation can be shown to be even more accurate when n a =. Note: In the following, all dotted curves will correspond to approximated data according to equation, whereas data obtained by monte-carlo simulations will be represented as continuous curves. Qx denotes the complementary error function under aussian statistics

3 a.... Fig. : β vs SN and IV. TADEOFF ANALYSIS L b MIMO In order to analyze the tradeoff between the packet errorrate ǫ and the transmission rate, when considering the optimization of the goodput, we first examine the relationship between ǫ and, and then consider the dependencyof with and with ǫ. A. ǫ versus The relationship between ǫ and is characterized by equations rigorous and approximated. Figure shows the curve ǫ versus for SN = db and {,,, }, in the, MIMO and MIMO cases. For, it can be noted that ǫ behaves like Q µ σ = for < µ, and like Q µ σ = for < µ, hence the step in the curve at absissa = µ in figure, for =. This result can also be analyzed by noting that lim I,n a, SN = µ is the channel ergodic capacity, C erg, and that strong channel coding ensures no error at the PHY layer as long as < C erg. For <, we see that the increase in ǫ is monotonous, whith an inflexion point at = µ. The only difference between the, MIMO and MIMO cases for the approximated curves reside in the absissa of the inflexion point. Otherwise, the curves look similar. For the exact curves, however, we can see that they are closer to the approximated curves in the MIMO case than in the case, which is consistent with the analysis of section III. B. Approximated expressions of and ǫ can be directly obtained by adequately mixing equations and. The curves are plotted in figures and respectively, both exact and approximated, for the, MIMO and MIMO channels.considerfirst the limit case, forwhich we have def = lim { = µ = µ, which, again, simply reflects the fact that error free transmission is achieved at the PHY layer when the transmission rate < C erg. Considering the dependency of with ǫ, we can also write: = µ ǫ. is thus linearly decreasing from µ = C erg when ǫ = to when ǫ =. In Fig. : versus ǫ, SN= db. MIMO x fact, due to the stepwise nature of the function ǫ versus when, only the two extreme points,µ and, in figure are achievable. For <, we can see on figure that for small values of. Then, as far as increases, errors appears at the PHY layer and thus <. achieves its maximum value at =, and then decreases up to when the PHY layer is no more reliable and consequentlythe numberof AQ retransmission tends to. Note that all curves coincide at the point µ, µ independently of, which is normal since at = µ we have ǫ =. Note also that we always have < µ. When comparing the behavior of for the MIMO and the channels, we observe that the distance between and µ is increasing with the number of antennas. In figure, which plots versus ǫ, we can also see that the value of ǫ that maximizes is smaller in the MIMO case. Hence, the PHY layer has to be made more reliable for a MIMO channel than for a channel when one attempts to optimize the goodput. V. OODPUT OPTIMIZATION Fromasystemlevelpointofview,we haveshowninsection IVthat it isimportanttomakethephy layerjust asreliableas necessary in order to maximize the goodput. For this purpose, we now examine how the optimal values ǫ = arg max ǫ = arg max evolve with the system parameters, n a and SN. Approximated values of and ǫ are easily obtained by derivation of the approximated expressions of : µ,na, SN Q σ,n a, SN ǫ µ,n a, SN σ,n a, SNQ ǫ ǫ

4 µ =.7.. µ =... µ =. 8 a 8 b MIMO Fig. : ǫ versus, SN = db. 8 c MIMO µ =.7... µ =. µ =. 7 8 a 8 b MIMO Fig. : versus, SN = db. 8 c MIMO whith respect to and ǫ. After some manipulation, we obtain that is solution of µ Q µ e σ = σ σ π and that and ǫ is solution of π ǫ e [Q ǫ ] +Q ǫ = µ σ where we have used µ = µ,n a, SN and σ = σ,n a, SN in order to make the equations clearer. Note than in the limit case we have, ǫ and µ,n a,sn = C erg. Figure presents the curve ǫ versus SN for n a {,,}and {,,}.We see that ǫ is decreasing with SN and. We also see that the valueof ǫ is quitelarge;for example,at db, ǫ >.. Furthermore,ǫ is much smaller in the MIMO case than in the case, and it tends to be less much smaller as SN and increases. Considering now and - whicharebothincreasingfunctionsof SN, and n a -, we present on figure 7 the ratios and C erg versus SN for n a {,,} and {,,,}. We see that these ratios are also increasing functions of SN, n a and, which means that the rate of increase of is The derivative of Q x is Q x = πexp [Q x]. higher than those of and C erg. This is due to the fact that there are two effects that contributes to the increase of = ǫ : the increase of and the decrease of ǫ. We conclude the discussion on figure 7 by noting that, as for given SN and the two ratios are increasing with n a, the AQ scheme under cross-layer control is more efficient in the MIMO case than in the case. In order to evaluate the impact on the goodput of a non optimal choice of ǫ, we present in figure 8 the ratio xǫ / def as a function of the SN, where xǫ = ǫ=x ǫ, for x =, n a {,,} and {,,,}. We see that the ratio is increasing with n a and, which means that MIMO and highly selective channels are more robust to a wrong choice of ǫ than and/or flat, slow-fading channels. VI. CONCLUSION In this paper, we have examined the properties of the natural tradeoff that exists in MIMO block-fading channels between the packet-errorrate at the PHY layer and the numberof AQ retransmissions, when rate adaptation is performed - based on statistical information about the channel - to optimize the goodput. The general findings are that the optimal packet errorrate at the PHY layer is smaller for MIMO channels than for channels - and that it diminishes with the number of antennas -, and that MIMO channels are less sensitive to a mismatch in the packet error-rate than channels. We also show that the AQ scheme under cross-layer control

5 .7. *. MIMO x *.. MIMO x * MIMO x.... a =. b = Fig. : ǫ as a function of the SN. c = * / * 8 * / * * / * * /C erg * /C erg * /C erg k= a b MIMO c MIMO Fig. 7: and C erg versus SN / / * / / * / / * 9 9 MIMO x 8 ǫ Fig. 8: versus SN for n a,, and {,,,} top: =, bottom: =. is more efficient in the MIMO case, in the sense that the achievable goodput is closer to the ergodic capacity than in the case. Some potentially interresting extensions of this work are considering a more interesting scenario where, first, the feedback channel introduces some errors, and, second a more sophisticated AQ scenario is envisaged. More over, it is interesting to extend this work to the case of multiple communicating pairs that interfer to each other, such as the MIMO multiple access channel. EFEENCES [] P. Wu and N. Jindal, Coding versus AQ in fading channels: how reliable should the phy be? in Proc. of IEEE LOBECOM, 9. []. T. S. Shakkottai, S. and P. C. arlsson, Cross-layer design for wireless networks, IEEE Communications Magazine, vol., pp. 7 8,. [] D. Chase, Class of algorithms for decoding block codes with channel measurment information, IEEE Transactions on Information Theory, vol. 8, pp. 7 8, 97. [] P. Wu and N. Jindal, Performance of hybrid-aq in block fading channels: A fixed outage probability analysis, IEEE Transactions on Communications, vol. 8, pp. 9,. [] A.. Chuang, A. uillen i Fabregas and I. L.. Collings, Optimal throughput-diversity-delay tradeoff in MIMO AQ block-fading channels, IEEE Transactions on Information Theory, vol. 9, pp , 8. [] C.. amal, H. E. and M. E. Damen, The MIMO AQ channel: diversity-multiplexing-delay tradeoff, IEEE Transactions on Information Theory, vol. 9, pp ,. [7]. L... i. F. A. Nguyen,. D. and N. Letzepis, MIMO AQ systems with multi-level feedback, in Proc. of ISIT. IEEE, June 9, pp. 8. [8]. i. F. A. Asyhari, A. T., Coding for the MIMO AQ block-fading channel with imperfect feedback and CSI, in Proc. of IEEE Information Theory Workshop ITW,.