Impact of Imperfect Channel State Information on ARQ Schemes over Rayleigh Fading Channels

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1 This full text paper was peer reviewed at the direction of IEEE Counications Society subject atter experts for publication in the IEEE ICC 9 proceedings Ipact of Iperfect Channel State Inforation on ARQ Schees over Rayleigh Fading Channels Le Cao, Pooi Yuen Ka and Meixia Tao, Dept. of Electrical & Coputer Engineering, National University of Singapore, Singapore 7576 Dept. of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China 4 Eails: caole@nus.edu.sg; elekapy@nus.edu.sg; xtao@ieee.org Abstract With iperfect channel state inforation (CSI) acquired by channel estiation at the receiver, the perforances of autoatic-repeat-request (ARQ) systes are evaluated as a function of the accuracy of channel estiation. A link between network-layer perforances and physical-layer paraeters is therefore established. We study in particular the goodput and the accepted packet error rate as a function of the channel estiation ean square error (MSE) and the factors which affect the MSE. The results enable us to analyze the optiu allocation of energy for data transission and energy for pilot channel estiation so as to axiize the goodput. I. INTRODUCTION Forward error correction (FEC) together with autoaticrepeat-request (ARQ) for retransission is coonly used to ensure the integrity of data transission. This is especially iportant for wireless counication where a deep fade in the channel can cause a oentary burst of errors in reception. Thus, uch work has been done on the perforance of FEC- ARQ schees over fading channels [] [9]. However, all these works, by and large, assue an AWGN channel [], [] or a fading channel with perfect channel state inforation (CSI) at the receiver [3] [9]. In practice, the CSI available at the receiver is acquired via channel estiation. The accuracy of channel estiation is liited by the aount of signal energy devoted to channel estiation, the channel noise and the teporal variations in the channel fading process. This eans that in practice the receiver CSI is usually iperfect, and it is iportant to be able to assess the ipact of the channel estiation error on the perforance of an FEC-ARQ schee. In this paper, we will deonstrate that the accuracy of the receiver CSI plays a crucial role in deterining the perforance of an FEC-ARQ syste. The perforance etrics we consider here are throughput/goodput and accepted packet error rate (APER). The accuracy of the receiver CSI is easured by the ean square error (MSE) of the channel estiates. The precise dependence of goodput and APER on the MSE is quantified. The results enable us to show in particular that when the MSE of channel estiation is above a certain critical value, the goodput drops very rapidly while the APER increases very fast. Below this critical value, the goodput increases very slowly and the APER also decreases ore gently. These results underscore the iportance of the The support of Singapore MoE AcRF Tier Grant T6B is acknowledged. MSs a perforance paraeter, and leads to the following very iportant syste perforance optiization proble. We assue that each packet is transitted with a certain fixed aount of total energy. Using a pilot-sybol-assisted channel estiation schee, part of this total energy is assigned to pilot sybols for channel estiation while the reainder is used for data transission. The question then is: what is the optiu fraction of the total energy that should be devoted to the pilot sybols for channel estiation? In this paper, we focus on establishing a link between network-layer perforances and physical-layer paraeters rather than analyzing the perforances of different ARQ schees. For the sake of siplicity, the selective repeat (SR)-ARQ syste is considered over block Rayleigh fading channels. The iperfect CSI is acquired via iniu ean square error (MMSE) channel estiation by aking use of pilot sybols. Two syste paraeters: APER P nd goodput η g [3] respectively are investigated. An upper bound on P E and a lower bound on η g are obtained in closed-for which is a function of the MSE of channel estiation. Furtherore, the aount of total power used for channel estiation is obtained, which can axiize the goodput of the syste. II. SYSTEM DESCRIPTION When inforation is transitted using an ARQ schee, each inforation strea with bits is sent to an error detection encoder. After passing through a binary (n, ) systeatic block encoder and being prefixed by N H pilot bits, a packet of bits is produced. Each packet coprises data bits for inforation transission, (n ) parity-check bits for error detection and N H pilot bits for channel estiation. All the bits of each packet are assued to be transitted by binary phase shift keying (BPSK) odulation. If buffering at both the transitter and the receiver is allowed, an SR-ARQ protocol can be ipleented. The transitter sends a continuous strea of packets. For each received packet, the channel gain is estiated using the pilot sybols. The received signals are then deodulated with the estiated channel gain and checked by the error detection code. When the receiver detects an error in a received packet, a retransission request for that packet is sent back to the transitter. The feedback channel is assued to be error-free. In this case the transitter, after a round-trip delay, responds to a retransission request by resending the requested packet /9/$5. 9 IEEE

2 This full text paper was peer reviewed at the direction of IEEE Counications Society subject atter experts for publication in the IEEE ICC 9 proceedings It then returns to the point at which it previously stopped and resues transission of new packets. The round-trip delay is assued to be larger than the coherence tie of the channel. Hence, the block fading gain experienced by the retransitted packet is independent of the gains experienced in previous transission(s) of the sae packet, and they are identically distributed. The block fading channel is assued to be constant over the duration of a packet. The channel gain h is a coplex Gaussian rando variable where R{h} and I{h} are independently and identically distributed (i.i.d) real Gaussian rando variables with eans zero and variances σ /. The received signal over the kth bit interval, is r p [k] = E p p [k]h + N[k], k =,,N H, () r s [k] = E s s [k]h + N[k], k = N H +,,N H + n, where E p and E s are the energy per pilot sybol and the energy per data sybol, respectively, p [k] and s [k] are the transitted pilot and data sybols, respectively. Ter N[k] is the zero-ean coplex AWGN with power spectral density N.TheN H pilot sybols in each packet are used in a Wiener filter for generating the MMSE estiate ĥ of h. Since the channel is coplex Gaussian, the MMSE estiate ĥ is given by [, eq.(.)] ĥ = N H i= w[i]r p [i], () where w[i] = σ E p (N H σ E p + N ) is the ith filter coefficient and is the sae for all i, since the channel gain h is constant. The channel estiation MSE is [, eq.(.49)] [ E h ĥ ] = V σ =. (3) σ +N E p H N The estiate ĥ is a coplex Gaussian rando variable with ean zero and variance σ V [, eq.(.8)]. III. ACCEPTED PACKET ERROR RATE AND GOODPUT ANALYSIS There are two basic paraeters fro which we can evaluate the perforance of an ARQ protocol: reliability and throughput. The throughput follows fro [, eq.(5-8)] [5, eq.(7)] as η = ( P d ), (4) where P d is the probability of detected error or the probability of retransission. The rate of the error detecting code is R = /n. The effective rate of each packet is defined as R e = /( ) which takes account of the redundancy introduced by the error detecting code and the pilot sybols used for channel estiation. The probability P d can be obtained by [, (Exaple5-)]. P d = P e P ue, (5) where P e is the probability that a received packet contains one or ore bit errors, and P ue represents the probability of the received packet containing an undetectable error pattern. The probability P e depends on the channel error statistics whereas the probabilities P d and P ue depend on both the channel error statistics and the choice of the (n, ) error detecting code. We first evaluate the probability P e. Based on the optial channel estiation receiver structure obtained in [, eq.(7)], the conditional bit error probability conditioned on the MMSE estiate of channel gain ĥ, is given by [, Appendix III] p = erfc E s ĥ E s V. (6) + N The energy per inforation bit at the input of the error detection encoder is Eb B = E s /R. Conditioned on knowing the estiated channel gain ĥ, the channel is eoryless since the AWGN is independent fro sybol to sybol. Hence, the conditional probability that a received packet contains at least one error bit, can be written as P e = ( p) n. (7) By averaging (7) over the Gaussian rando variable ĥ, the packet error probability P e can be obtained as P e = ( ( p) n ) l= σ V e ĥ /(σ V ) d ĥ. (8) In Appendix I, we use the Chernoff bound to show that P e is tightly upper-bounded by P e (b +(j +)c), (9) j= where, b = σ V, c = E s E s V. () + N Next, we evaluate the probability of undetectable error P ue. For (n, ) linear codes, except for soe short linear codes, the weight distributions for any linear codes are still unknown. Consequently, it is considerably difficult to copute their P ue, but it is fairly easy to derive an upper bound for the enseble of all (n, ) linear codes. Conditioned on knowing the estiated channel gain ĥ, the upper bound on the undetectable error probability can be evaluated by [, eq.(3.4)] P ue (n ) [ ( p) n ]. () Taking the ean of () over ĥ, the undetectable error probability is upper-bounded as P ue (n ) [ ( p) n ] σ V e ĥ /(σ V ) d ĥ. () The last integral can be evaluated like that in (8), giving P ue (n ). (b +(j +)c) l= j= (3)

3 This full text paper was peer reviewed at the direction of IEEE Counications Society subject atter experts for publication in the IEEE ICC 9 proceedings The ratio between P ue and P e can then be upper bounded by the ratio of () to (8): P ue P e (n ). (4) We thus have P ue << P e, and therefore P d P e fro (5) when the nuber of parity-check bits n is larger than. The throughput is then lower bounded by η η L = ( P e ). (5) One can only evaluate a lower bound to right-hand side of (5) by using the upper bound on P e in (9). Another useful syste paraeter, the APER, which shows the reliability of the ARQ syste, is given by [, eq.(5-)] P E = P ue. (6) P d Using P d P e and substituting (3) and (9) into (6), P E is upper-bounded by P E (n ) ( Z), (7) Z where, (b +(j +)c). (8) l= j= The throughput is eaningful only when considered in conjunction with the reliability. Therefore, the goodput η g is defined as the ratio of the expected nuber of inforation bits correctly received per unit of tie to the total nuber of bits that can be transitted per unit of tie [3]. The goodput, which shows the proportion of the throughput consisting of correct packets, can be expressed as η g =( P E )η. (9) By substituting (6) and (4) into (9), the goodput can be obtained as η g = ( P e ). () The goodput can be considered as a lower bound on the throughput, since the η g in () is the sae as the η L in (5). Substituting (9) into (), we obtain the lower bound on η g as η g (b +(j +)c). () l= j= IV. POWER ALLOCATION BETWEEN PILOT AND DATA BITS Each packet is sent with a fixed total energy E T. When ore energy is devoted to channel estiation, the estiates of channel gains are ore accurate, leading to a saller error probability. However, this reduces the energy available for data transission and leads to a higher error probability. For this reason, there ust exist an optiu fraction ε of the total energy E T that should be devoted to channel estiation so as to axiize the lower bound on the goodput η g in (). All signalling essages are assued to be significantly shorter than the user data packets, and therefore transitted with negligible overall energy consuption. For a given total packet energy E T, the aount of total energy assigned to pilot sybols is εe T = N H E p while the reainder of total energy devoted to data transission equals ( ε)e T = ne s.we assue N H to be fixed for each packet. Thus an optiu ε will lead to an optiu E p and siilarly an optiu E s. Since N H is fixed, () shows that axiizing the goodput η g aounts to iniizing the packet error rate P e.usingthe lower bound (), the axiization proble now coes to ( c ) ε = arg ax ε f = b l= j= where b and c given in () can be rewritten as and c = b = σ Nσ N +εe T σ = +εγ εσ γ (b +(j +)c), () (3) ( + εγ)( ε)e T ( ε)e T σ + n(n + εe T σ ). (4) Here, γ = E T σ /N is defined as the total transit signal-tonoise ratio (SNR). The objective function f ( c b) is a onotonically increasing function of the variable c/b. Therefore, the optiization proble can be reduced to { c } { ε ε( ε)γ } = arg ax = arg ax. ε b ε γ( ε)+n( + εγ) (5) Setting the derivative of c/b with respective to ε equal to zero, and solving the resulting quadratic equation, we obtain the optial ε as ε = n + γ n + nγ + γ n + γn. (6) γ nγ The optial ε, which is in the range [, ], satisfies an equality shown in the following proposition. The proof is given in Appendix II. Proposition 4.: For a given n, the optial value of ε satisfies the following inequality for any γ. n n ε.5 (7) This proposition indicates that the optiu aount of energy devoted to channel estiation is always less than half of the total energy. On the other hand, at least a fraction of total energy (i.e. n n E T ) is necessary to be devoted to channel estiation. A sall aount of energy is assigned to channel estiation when a long code (i.e. large n )isused.thisis because the left-hand side of (7) can be reduced to / n when n is large.

4 This full text paper was peer reviewed at the direction of IEEE Counications Society subject atter experts for publication in the IEEE ICC 9 proceedings 5 R=.9 R=.88 R=.84 5 R=.9 R=.88 R=.84 Lower Bound on Goodput η g R=.9 R=.88 R=.84 Lower Bound on Goodput η g /N =3 db /N =5 db R=.9 R=.88 R=.84 /N =5 db Noralized MSE (V /σ ) 5 3 Average SNR per Bit /N 4 Noralized MSE (V /σ ) Fig.. The APER versus the noralized MSnd /N. Fig.. The goodput versus the noralized MSnd the APER. V. NUMERICAL RESULTS Since the bounds on goodput and APER only depend on the values of n and and they are not influenced by specific code structures, different values of n and are chosen to deonstrate the perforances. The fixed total energy E T for each packet is ( + N H ), where represents the average energy per bit. The noralized channel estiation MSE V /σ depends on the energy ε E T devoted to pilot sybols rather than the value of N H. Without loss of generality, the nuber of pilot sybols is assued to be N H =5. In Fig. and Fig., the perforance bounds are plotted with the nuber of inforation bits = and with an equal power allocation E p = E B b = (i.e. ε = N H /(n + N H )). Fig. indicates that there exists a critical value for both V /σ and average SNR per bit /N that separates two different trends in the APER curves. In particular, when V /σ is above a critical value of around,theaper deteriorates very fast. Below this critical value, the APER decreases ore gently. The corresponding critical value of SNR /N is around db. Additionally, it can be seen that in order to achieve a certain APER requireent, greater channel estiation accuracy or higher /N is needed for a code with fewer parity-check bits. Fig. shows how the paraeter V /σ and the APER affect the goodput perforance. The values of noralized MSbove a critical value of around degrade the goodput perforance very rapidly. The goodput can be iproved by ore accurate channel estiation. However, decreasing the noralized MSE below a value of about 3 leads to diinishing increents in the goodput. Furtherore, in order to achieve a certain goodput requireent, greater channel estiation accuracy is needed for a code with ore parity-check bits. This is because using ore accurate channel estiation copensates for the higher packet error rate induced by a longer packet. The goodput versus the APER in Fig. shows that, for a lower-rate code, a saller APER is required to achieve desired goodput perforance. For afixedsnr /N, higher goodput is obtained by using a higher-rate code but at the expense of worse APER. The above discussion of channel estiation accuracy leads to the consideration of optiu power allocation between channel estiation and data transission to achieve the axiu goodput. The power allocation and the goodput iproveent achieved are shown in Fig. 3 and Fig. 4, respectively. The curves in Fig. 3 indicate that ε is a decreasing function of the total transit SNR γ but it converges to n n and.5 in the high and low SNR regions, respectively. For different codes, an iproveent of about.5 db to db can be observed in Fig. 4 for a given goodput. Less iproveent is achieved when a longer code is used. For a code with = and R =.84, a.5 db iproveent is obtained. VI. CONCLUSION With iperfect CSI, we have studied the effect of channel estiation error on the perforance of an SR-ARQ syste over block Rayleigh fading channels. Closed-for bounds on APER and goodput have quantified the perforance iproveent due to ore accurate CSI. A power allocation between pilot bits and inforation bits has been obtained for achieving axiu goodput. The percentage ε of total power assigned to pilot bits is a decreasing function of total transit SNR γ, but converges to.5 and n n in the low and high SNR regions, respectively. For the sake of siplicity, only the SR- ARQ schee without channel coding is considered as an exaple to deonstrate the effects of iperfect CSI. In our future work, ore advanced channel coding and ARQ schees will be considered to study the effects of iperfect CSI. APPENDIX I By using the Chernoff bound: erfc(x) <e x, an upper bound can be obtained as ( P e ) n e c ĥ e b ĥ d(b ĥ )= Z.

5 This full text paper was peer reviewed at the direction of IEEE Counications Society subject atter experts for publication in the IEEE ICC 9 proceedings R=.83, = R=.9, = R=.5, =5. R=.84, = Average SNR per Bit /N Average SNR per Bit /N ε * ε * Lower Bound on Goodput Fig. 3. The value of ε versus average SNR per bit /N.9 ε=n.8 H /(n+n H ) ε=ε *.7 R=.9, =.6 R=.84, =.5.4 R=.83, =5.3 R=.5, =5.. Note that when l =,ter l j= APPENDIX II (n j)c (b+(j+)c) is defined to be. The optial value ε is a onotonically decreasing function of γ, which can be shown as follows. By changing to the variable x = γ n, (6) becoes ε = +x +x + nx + nx. (3) x nx By taking the first derivative of ε with respective to x and siplifying, the derivative can be arranged to be dε dx ( (n ) +x + nx + nx (nx + x +) ) = +x + nx + nx (x nx). By applying the inequality of arithetic and geoetric eans, the inequality dε /dx is true because of the relationship +x + nx + nx = ( + x)(nx +) +nx + x. Using the onotonically decreasing property of ε, the upper and lower liits of ε can be seen to be li x ε (x) ε li ε (x). (3) x The lower liit can be shown to be +x x li + x + nx + n = n x n n. By applying L Hospital rule, the upper liit is obtained as.5( + n +nx)( + nx + nx + x) li x n =.5 Fig Average SNR per Bit /N The lower bound on the goodput achieved by different values of ε Using integration by parts with u =( ) n and dv = e c ĥ de b ĥ, [uv vdu] can be expressed as ( ) n + nc ( e c ĥ ) n e ĥ (b+c) d ĥ (8) Continuing the integration by parts with u =( e c ĥ ) n and dv = e ĥ (b+c) d ĥ, and perforing the siilar process till the last integral, ter Z coes to ) n + nc ( ) n nc(n )c + ( + Hence, we can get ( ) n (b + c) (b + c)(b +c) nc(n )c c (b + c)(b +c) (b + nc). (9) l= j= (b +(j +)c). (3) REFERENCES [] S. B. Wicker, Error Control Syste for Digital Counication and Storage, st ed. Upper Saddle River,New Jersey: Prentice-Hall, 995. [] S. Lin and D. J. Costello, Error control coding. Englewood Cliffs, NJ: Prentice-Hall, 4. [3] D. Qiao, S. Choi, and K. G. Shin, Goodput Analysis and Link Adaptation for IEEE 8.a Wireless LANs, IEEE Tansaction on Mobile Coputing, vol.,. [4] J. Yun and M. Kavehrad, Markov Error structure for Throughput Analysis of Adaptive Modulation Systes Cobined with ARQ over Correlated Fading Channels, IEEE Transaction on Vehicular Technology, vol. 54, 5. [5] Y. Zhou and J. Wang, Optiu subpacket transission for hybrid ARQ systes, IEEE Transaction on Counications, May 6. [6] J.-F. T. Cheng, Coding Perforance of Hybrid ARQ Schees, IEEE Transactions on Counications, vol. 54, no. 6, 6. [7] H.-C. Yang and S. Sasankan, Analysis of Channel-Adaptive Packet Transission Over Fading Channels With Transit Buffer Manageent, IEEE Transactions on Vehicular Technology, vol. 57, no., 8. [8] A. Steiner and S. S. (Shitz), Multi-Layer Broadcasting Hybrid-ARQ Strategies for Block Fading Channels, IEEE Transactions on Wireless Counications, vol. 7, no. 7, 8. [9] C. Pientel and R. L. Siqueira, Analysis of the Go-Back-N Protocol on Finite-State Markov Rician Fading Channels, IEEE Transactions on Wireless Counications, vol. 57, no. 4, 8. [] S. Haykin, Adaptive Filter Theory, 4th ed. Upper Saddle River,New Jersey: Prentice-Hall,. [] P. Y. Ka, Optial Detection of Digital Data Over the Nonselective Rayleigh Fading Channel with Diversity Reception, IEEE Transactions on Counications, vol. 39, no., pp. 4 9, Feb 99.