On the Average Rate Performance of Hybrid-ARQ in Quasi-Static Fading Channels

Transcription

1 1 On the Average Rate Performance of Hybrid-ARQ in Quasi-Static Fading Channels Cong Shen, Student Member, IEEE, Tie Liu, Member, IEEE, and Michael P. Fitz, Senior Member, IEEE Abstract The problem of efficient communication over a scalar quasi-static wireless fading channel is considered in this paper. The shortcomings of two well-known schemes, Single-Layer Transmission (SLT) and Multi-Layer Transmission (MLT), are pointed out. It is shown that these disadvantages can be eliminated when Hybrid-ARQ (HARQ) is used, provided that the rate assignment is carefully done. The average rate performance of several HARQ schemes is optimized and compared. In addition, optimal power allocation between the primary transmission and the ARQ retransmission is derived and shown to further increase the average rate. This power allocation gain is remarkable at low SNR, but becomes negligible at high SNR. Comparison of two limited feedback communication schemes, ARQ feedback and quantized CSI feedback, is discussed from several perspectives. Although the optimization problem is formed with respect to the average rate performance, simulation results give a comprehensive comparison of these schemes under different metrics, including average rate, outage probability, and the combination of both. Substantial performance improvement is observed with even one ARQ retransmission in all simulations. Most importantly, this gain seems to be robust with respect to the fading distributions. I. INTRODUCTION The problem studied in this paper is how to efficiently transmit information over a quasi-static wireless fading channel, where the channel gain is constant during one transmission block and changes independently from one block to another. It is assumed that the receiver can perfectly track the fading process, The work of Cong Shen and Michael Fitz is supported by NSF grant CCF , and by ST Microelectronics. Part of this work was done while Cong Shen was visiting Texas A&M University. Cong Shen and Michael P. Fitz are with the Department of Electrical Engineering, University of California Los Angeles (UCLA), Los Angeles, CA 90095, USA. {congshen,fitz}@ee.ucla.edu. Tie Liu is with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843, USA. tieliu@ece.tamu.edu.

2 2 but the transmitter has no knowledge of the channel realization other than the statistical characterization. For this communications scenario, the Shannon capacity is zero as there is always a nonzero probability that the channel is in a deep fade. A useful and well-accepted performance metric is the outage capacity [1]. In this formulation, a fixed-rate channel code is used, and the information is reliably transmitted if the instantaneous channel gain supports the predetermined transmission rate. Otherwise, an outage is declared, and no information can be recovered at the receiver. Quasi-static fading with channel state information (CSI) available only to the receiver (CSIR) is a major display of the detrimental effect of fading. Recently, average rate has been proposed as an alternative performance measure for communication over quasi-static fading channels [2], [3]. Different names of the same concept are scattered in the literature, including throughput 1 in [5], expected rate in [6], and expected capacity in [7]. In this paper, the term average rate is adopted. The main idea behind this metric is an opportunistic view of fading: a well-designed system should be able to adapt to the channel fading, i.e., it sends some information across the channel when the channel is not-so-good and a lot of information when the channel is very good. By exploiting the good channel realizations, the long term average of the reliable information rate, i.e., the average rate, can be substantially improved. Note the sharp contrast between the opportunistic view of fading and the nonadaptive transmission strategy used in the outage capacity formulation. If the transmitter has access to CSI, it can optimally adjust the transmission rate and power based on the instantaneous channel gain. Research along this line has been well pursued, e.g., [8] for singleuser fading channel and [9] [11] for multi-user fading channel. Adapting to channel fading without transmitter s knowledge of CSI, however, faces some conceptual difficulties. The breakthrough was made by Cover in [12], where he observed the similarity between communication over quasi-static fading channels and broadcasting to multiple users. This venue was later pursued by Shamai [2], Shamai and Steiner [3], and Liu etc. [5], leading to the development of a multi-layer transmission (MLT) strategy which utilizes broadcast superposition coding. By organizing information into layers, the MLT strategy allows rate adaptation to channel fading at the expense of creating self interference during the decoding of the earlier-decoded layers in the stack. Furthermore, depending on the actual channel realization, it may occur that only part of the information can be decoded at the receiver, which may lead to some extra complications in the upper layers in the network hierarchy. 1 Note this is not the same throughput concepts that is widely used in the ARQ literature [4]. A comparison is made in Section V.

3 3 This work follows the same line of examining the average rate performance of communication over quasi-static wireless fading channels [2], [3], [5]. The shortcomings of both Single-Layer Transmission (SLT) and MLT are first pointed out. It is then shown that with the use of Hybrid-ARQ (HARQ), all these disadvantages can be totally eliminated, provided that the rate assignment of the HARQ protocol is carefully designed. This scheme is also referred as Rate-Optimized HARQ (RO-HARQ). The basic idea is to exploit the existence of ARQ in the data link layer to increase the average rate. Roughly speaking, the initial transmission is set to be very aggressive (high rate). If the channel does not support this high rate, an ARQ will help by indicating the transmitter to reduce the rate. The average rate maximization problem of RO-HARQ is formulated, and numerical results demonstrate the remarkable gain over both SLT and MLT strategies. Moreover, this gain is robust to the fading distributions. Further average rate increase is possible, especially in the low signal-to-noise ratio (SNR) regime, if power allocation between the primary transmission and ARQ retransmission is allowed. To comprehensively compare these schemes, computer optimization and simulations are performed with respect to different performance measures, including average rate, outage probability, and the combination of both. Such numerical comparison demonstrates the gain of RO-HARQ over SLT and MLT from different perspectives. Using ARQ to improve performance is not a novel idea. In fact, HARQ techniques are widely used in most of the current digital communication systems. A good summary of the progress of HARQ schemes is presented in [13]. Early work regarding the HARQ system focuses on the usage of algebraic error-correction and error-detection codes [4]. Recent interests of HARQ mainly originate from the rapid progress of wireless communications, where high-rate reliable transmission faces the challenge of severe channel fluctuations. An information-theoretic throughput and delay analysis of several HARQ schemes in the Gaussian collision channel is reported in [14]. Throughput analysis of incremental redundancy HARQ in the block-fading AWGN channel is carried out in [15]. From the practical implementation point of view, research interests have shifted from traditional algebraic linear block codes to the more powerful capacity-approaching modern codes. For example, the problem of designing Low-Density Parity-Check (LDPC) codes for the HARQ protocol has been addressed in [15] [20]. However, in almost all the existing literature, the performance metrics of HARQ are either the packet error rate, delay, or throughput. The recently developed information-theoretic metric, average rate, indicates the mean value of the successful transmission rate and should also be considered as a performance measure of HARQ. This is another motivation of the work. The rest of this paper is organized as follows. Section II defines the system model. Section III briefly discusses the average rate performance of SLT and MLT. Section IV presents the theoretical analysis of

4 4 RO-HARQ. It first points out the problems of the known schemes, and then analyzes the RO-HARQ scheme. Average rate maximization is discussed in Section IV-C and IV-D, followed by the optimal power allocation in Section IV-E. Numerical comparison with several different performance metrics is reported in Section V. Section VI discusses the difference between ARQ feedback and quantized CSI feedback. Finally, Section VII concludes the paper. II. SYSTEM MODEL Assume that the channel is stationary. Consider the following scalar quasi-static wireless fading channel: y[m] = h[m]x[m] + z[m], m = 1,, L (1) where the channel gain h[m] = h is random but remains constant during the entire transmission, {x[m], m = 1,, L} is a length-l codeword, and {z[m]} is independent and identically distributed (i.i.d.) complex Gaussian distributed noise with zero mean and variance N 0 : z CN (0, N 0 ). It is assumed that L is large enough so that coding can achieve the channel capacity. There is a short-term average power constraint of P on {x[m]}. This constraint prohibits power allocation across different fading states 2, which has been studied in [8]. The channel gain and noise power are normalized to be E [ h 2] = 1 and N 0 = 1 respectively, so the average received SNR is SNR =. P E [ h 2] /N 0 = P. The channel random gain g =. h 2 0 is assumed to be a continuous variable with the cumulative distribution function (CDF) F (g) and the probability density function (PDF) f(g). One such distribution is the frequently-encountered Rayleigh fading with h CN (0, 1). The discussion in this work applies to general fading distributions. The quasi-static fading channel is a good model for users that are stationary, or moving slowly relative to the rate of communication. Due to the slowly-varying nature of the channel, channel estimation at the receiver can be performed with high accuracy. Thus, it is reasonable to assume perfect CSI at the receiver. This assumption will be made throughout this paper. The focus of this work is on high-rate, delay insensitive applications such as data traffic in wireless LAN. For such applications the use of capacity-approaching channel codes with long block length can be justified, and information-theoretic results are good approximations of real-world performance. This motivates to take an information-theoretic view and consider the capacity-related measure on the system performance, as is done in this paper. 2 When discussing power allocation for RO-HARQ in Section IV-E, unequal power allocation between the primary transmission and ARQ retransmissions is allowed. Notice that even in this case there is no power allocation across different fading states.

5 5 III. AVERAGE RATE OPTIMIZATION OF SINGLE-LAYER AND MULTI-LAYER TRANSMISSIONS The average rate of a scheme is defined as the long term average of the reliable communication rate: R =. E g [R(g)] (2) where g is the channel state and R(g) is the successful transmission rate when channel is g. See [7] for a formal definition. It should be noted that this definition of average rate is different from the ergodic capacity; here coding across different fading states is prohibited. As another note, this definition of average rate is also different from the typical definition of throughput in ARQ literature, which has been intensively studied in [14], [15], [21] [23], to name a few. These results for throughput do not apply in our work. In Section V, a comparison of these two ARQ metrics will be made. A. Single-Layer Transmission (SLT) Assuming that the transmitter does not know the channel realization h, a simple and well-adopted scheme for communication over a slow fading channel is to send the data at a fixed rate R nats per channel use (npcu) 3. If the instantaneous channel realization supports the rate R, the receiver gets a successful transmission; otherwise, it declares an outage. In this scheme, the random variable R(g) is R, if log (1 + gp ) R R(g) = 0, otherwise and the average rate can be calculated as (3) R SLT = R P {log (1 + gp ) R} ( ( e R )) 1 = R 1 F. (4) P In order to maximize the average rate, one needs to solve the following optimization problem: ( )) maximize R 1 F subject to R 0. ( er 1 P It is easy to check that the objective function is in general not concave. However, since the function is differentiable and upper bounded (by the ergodic capacity), the optimal solution exists and must satisfy the standard Karush-Kuhn-Tucker (KKT) condition: ( e R ) ( 1 1 F ReR e R ) P P f 1 = 0. (6) P (5) 3 The information unit is nat throughout the paper, except for the numerical results.

6 6 With Rayleigh fading, the KKT condition simplifies to Re R = P RSLT,Rayleigh = L(P ) (7) where L(x) is the Lambert W function [24], and ( ) R SLT,Rayleigh = L(P ) exp el(p ) 1. (8) P B. Multi-Layer Transmission (MLT) This scheme adopts the multi-user broadcast superposition code for a single-user slow fading channel. It was proposed in [3] for infinitely-many layers, and in [5] for finite layers. In MLT, it is still assumed that the transmitter has no knowledge of the fading realization h. Let us focus on N = 2, as has been pointed out in [5], 2-layer superposition coding already achieves most of the average rate gain in a Rayleigh slow fading channel. In this case, the transmitter splits the message w into two sub-messages w 1 and w 2. They are separately encoded into x 1 and x 2 respectively, assuming Gaussian signaling, and finally superposed into the transmit signal x: x = x 1 + x 2 (9) where x 1 CN (0, αp ) with rate R 1 npcu, x 2 CN (0, (1 α)p ) with rate R 2 npcu, and the scalar α [0, 1] determines the power allocation between different layers. At the receiver, the decoding is carried out based on the channel realization h. The first step is trying to decode x 1, treating x 2 as part of the noise. If x 1 is not successfully decoded, the receiver will give up and declare an error. On the other hand, if x 1 is successfully decoded, the decoding procedure continues to the second step: subtracting the successfully decoded x 1 and then decoding x 2. After some manipulation, the 2-layer MLT average rate maximization problem reduces to: where s 1. = e R 1 1 P (1 (1 α)e R 1) and s 2 maximize R 1 (1 F (s 1 )) + R 2 (1 F (s 2 )) subject to R 1 0 R 2 0 solution exists and can be numerically calculated. (10) e R 1 +R 2 e R 2 e R 1 +R 2 1 α 1 =. er 2 1 P (1 α). With the same argument as in Section III-A, the optimal The average rate of MLT can be further boosted by increasing the layers of the superposition code, although such gain has been shown not to be significant in Rayleigh fading [5]. In [3], Shamai and

7 7 Steiner derived the maximum average rate of a superposition code with infinitely-many layers for the Rayleigh fading: where S 0 = 2/ ( P ) and E i (x) = x R MLT,Rayleigh = 2E i(s 0 ) 2E i (1) ( e S0 e 1) (11) e t t dt, x 0 is the exponential integral function. A. Motivation IV. RO-HARQ: THEORETICAL ANALYSIS The development of Rate-Optimized Hybrid-ARQ (RO-HARQ) is motivated by the disadvantages of SLT and MLT, which are presented in this section. The SLT scheme evaluates the transmission in an on-off fashion: the transmission is either entirely successful or totally failed. This scheme matches the concept of capacity versus outage, but suffers from the following two major disadvantages when used in a slow fading wireless channel. 1) It cannot send any information in a bad channel realization. As the instantaneous channel gain drops below a threshold, no information can be guaranteed to get decoded at the receiver. A better scheme should allow some information to be successfully decoded even when the channel is bad. 2) It cannot make full use of a good channel realization. This is the other extreme of the random fading gain. If the instantaneous channel gain is very high, theoretically much higher data rate can be successfully transmitted. However, the SLT scheme wastes such good channel gains by using a fixed-rate code for transmission. To summarize, the SLT scheme uses a nonadaptive transmission strategy which suffers from both over-utilizing and under-utilizing the channel. The MLT scheme is mainly devised to overcome the disadvantages of the SLT scheme without transmitter s knowledge of CSI. The main idea is to organize the information into layers at the transmitter. The receiver, on the other hand, can choose to decode some of the layers based on the actual channel realization. Organizing information into layers incurs an additional cost: it creates self interference to the decoding, which does not exist in the SLT scheme. To be specific, when the receiver decodes the n-th layer, it treats all the not-yet-decoded layers n+1,, N as interference and thus decreases the effective receive SNR: SNR n = gp n 1 + g N i=n+1 P. i As the number of layers increases, this self interference affects more layers in the decoding; in fact, only the last decoded layer is interference-free. Although one can manipulate the power allocated to different

8 8 layers to alleviate this issue, there is a tension in doing so: giving more power to one layer will create stronger interference to other previously decoded layers; giving less power to one layer will reduce the decodable rate of this layer. Thus, power allocation cannot totally eliminate the self interference. B. The RO-HARQ Scheme This paper focuses on the HARQ scheme with optimally designed rate for each transmission. It will be shown that this scheme can eliminate all of the aforementioned disadvantages with optimized rate assignment. With the help of HARQ, the decoding status at the receiver will be reported back to the transmitter via the ARQ feedback, which indicates the successful decoding of received signal by acknowledgement (ACK) and failed decoding by negative acknowledgement (NACK). It is assumed that the ARQ feedback channel is delay-free and error-free. The maximum number of retransmissions of HARQ is denoted by N, i.e., the total transmissions (including the initial one) cannot exceed N + 1. In the ARQ literature, N + 1 is also called the maximum allowable ARQ rounds. The choice of N reflects the worse-case delay caused by the ARQ retransmissions, and is generally determined by the system delay requirement. For example, choosing small N models certain delay critical situation. Note that the HARQ protocol is typically available in almost all wireless systems, and thus exploiting HARQ will not need additional designs. Define K b to be the number of information nats that are planned to be transmitted with HARQ. We consider to optimize three HARQ protocols [14], [23] in this paper. 1) ALO. The transmitter encodes the K b information nats into a codeword of length L with rate R = K b /L npcu, and then keeps sending the same encoded packet in every retransmission. The receiver only decodes the most recently received packet. This loop goes until the ACK is declared by the receiver, or the maximum retransmissions are used. 2) RTD. The transmitter is the same as ALO. The receiver performs a maximum ratio combining of all the received packets. This loop goes until the ACK is declared by the receiver, or the maximum retransmissions are used. 3) INR. The transmitter encodes the K b information nats into a codeword of length L. Then it serially punctures the length-l codeword into N +1 sub-codewords with strictly decreasing rates Kb L 1 > Kb L 2 > > Kb L N+1. The truncation length of each sub-codeword is the design parameter. It is determined by the code rate optimization and will be addressed in Section IV-C. At the n-th transmission

9 9 (n = 1,, N), the transmitter reduces the total rate to 4 R 1 + R R N+2 n = Kb L n by sending additional redundancy symbols such that the overall transmitted sub-codeword length is L n. The receiver tries to decode all the data it receives up until this moment. This loop goes until the ACK is declared by the receiver, or the maximum retransmissions are used. At the last retransmission (step N + 1), rate R 1 is tried without any ARQ feedback. If decoding is still unsuccessful, a decoding failure is declared. This works focuses on INR due to the following two reasons. 1) ALO scheme does not work in the slow fading channel. The reason is that there is no time diversity to exploit, as the channel gain remains constant over ARQ retransmissions. Thus keep sending the same packet and only decoding the most recently received one will not increase the chance of being successfully decoded. 2) RTD has an average rate performance that is inferior to INR. This will be shown both theoretically and numerically in the next section. Note although RTD has inferior performance, it has some other practical advantages which make it attractive. For example, in RTD the retransmitted packet is always the same, which is easier to do than INR, in which additional parity nats have to be generated and transmitted. As another example, the same retransmitted packet in RTD makes the packetization simple, while in INR the length of the retransmitted parity nats may vary from one retransmission to another. The novelty of the proposed RO-HARQ scheme is the optimized rate assignment 5 of each transmission. Traditionally, HARQ is used in a passive manner in a wireless system. The purpose of HARQ is to indicate the decoding status to the transmitter, such that the transmitter can save the bad packet by retransmitting. The existence of HARQ is typically not exploited by the transmitter; it is treated only as a binary indicator of the decoding status, and other modules in the system (e.g., channel coding, modulation, etc.) are operating as if HARQ were not available. Due to this reason, the rate of each HARQ transmission and the (equal) power allocation is usually predetermined. This work proposes the different view that HARQ can be exploited by the transmitter to provide better average rate and outage performance, i.e., it can be utilized in an active manner. This new view suggests that the rate and power 4 The reason of using this seemingly redundant expression for the data rate is that it simplifies equations (12), (13), (14), (15), and the proof of Theorem 2. 5 Another novelty is the optimal power allocation among transmissions to further improve the average rate performance. This is discussed in Section IV-E.

10 10 associated with each HARQ transmission can be optimized according to the channel fading statistics to achieve better performance. The main idea is that since the transmitter is aware of the HARQ link in the system, it can transmit very aggressively (at very high rate) even if it does not know the random channel gain g before transmitting. Then if the channel is not good, this high-rate transmission will fail, and the HARQ can save it by indicating this failure to the transmitter so that it can adjust to a lower rate. On the other hand, if it is lucky that the channel is strong enough, such gambling will bring high return: the strong channel realization is (almost) fully utilized, i.e., there is (almost) no waste of the good channel. This is the basic idea behind RO-HARQ. It is now clear why RO-HARQ can eliminate the shortcomings of SLT and MLT. By transmitting aggressively, the good channel realizations are fully used; with the ARQ feedback, the transmission rate can be reduced for the bad channel realizations. At the same time, as there is no superposition or layering in the channel coding, self-interference does not exist. C. Average Rate Maximization of INR and RTD The INR scheme is described in the previous section. With INR, the random variable R(g) is N+1 i=1 R i, if log (1 + gp ) N+1 i=1 R i R(g) = n i=1 R i, if n+1 i=1 R i > log (1 + gp ) n i=1 R i, n = 1,, N 0, if R 1 > log (1 + gp ) The average rate of INR can thus be computed as (12) where g n R N INR = (R R N+1 ) P {log (1 + gp ) R R N+1 } = = + (R R N ) P {R R N+1 > log (1 + gp ) R R N } + + (R 1 + R 2 ) P {R 1 + R 2 + R 3 > log (1 + gp ) R 1 + R 2 } + R 1 P {R 1 + R 2 > log (1 + gp ) R 1 } { } n R n P log (1 + gp ) R k N+1 n=1 N+1 n=1. = e n k=1 R k 1 P. k=1 (13) (14) R n (1 F (g n )) (15) The problem is to choose the non-negative design parameters {R 1,, R N+1 } to optimize the average rate (15). Clearly, the non-negativity constraints on {R n } can be dropped, and the INR average rate

11 11 maximization problem becomes maximize N+1 n=1 R n (1 F (g n )). (16) A direct evaluation of the KKT condition gives for all n = 1,, N, and 1 F (g n ) R ne n k=1 Rk P f (g n ) N+1 i=n+1 1 F (g N+1 ) R N+1e N+1 k=1 Rk P R i e i k=1 Rk In some simple case (e.g., small N), these equations can be numerically solved. P f (g i ) = 0 (17) f (g N+1 ) = 0. (18) The average rate of RTD can be similarly computed. In the n-th transmission of RTD, n = 1,, N +1, the receiver performs maximum ratio combining of the n received packets. This effectively increases the receive SNR to SNR n = np, while reducing the data rate to R/n. Thus the random variable R(g) is R n, if log (1 + ngp ) R > log (1 + (n 1)gP ), n = 1,, N + 1 R(g) = (19) 0, if R > log (1 + (N + 1)gP ) The average rate can be computed as where q n problem is R N RT D = = N+1 n=1 N+1 n=1 R n P {q n 1 > g q n } (20) R n (F (q n 1) F (q n )) (21). = e R 1 np for n = 1,, N + 1, and q 0. =. Similarly, the RTD average rate maximization maximize N+1 n=1 R n (F (q n 1) F (q n )). (22) Numerical evaluation and comparison of the optimal average rate of (16) and (22) is given in Section V, where it will be seen that RTD is worse than INR. In fact, this can be analytically proved: Theorem 1: Denote the optimal solution to problem (16) as ({R1,, R N+1 }, R N INR ), and the optimal solution to problem (22) as (R, R N RT D ). Then Proof: See Appendix A. R N RT D R N INR (23) In order to fully enjoy the benefits of RO-HARQ, the channel coding design has to satisfy the following two requirements.

12 12 1) A good mother code that can be serially punctured into several different optimized code rates. 2) A single decoder that can handle all the rates such that the coding performance of each sub-code is capacity-approaching. In fact, there are several existing code designs, developed in both academia (e.g., [15], [17], [18], [20], [25]) and industry (e.g., [26]), that satisfy most of these requirements. D. Asymptotic Average Rate of INR It is of interest to ask what is the asymptotic performance limit of the INR scheme (by allowing N goes to infinity, i.e., infinitely many retransmissions). Such asymptotic performance gives the ultimate limit of the proposed scheme. The following theorem states that the average rate of INR asymptotically converges to the ergodic capacity, even though coding across fading blocks is prohibited in INR. Theorem 2: As N, the average rate of INR with equal power allocation among transmissions converges to the ergodic capacity of the fading channel. Proof: The average rate of an INR-ARQ scheme with a given N is given in equation (14) as { } N+1 n R INR N = R n P log (1 + gp ) R k. n=1 As N, this average rate converges to R INR = where equation (24) is due to the fact that for a non-negative random variable X. 0 k=1 P {log (1 + gp ) R} dr = E [log (1 + gp )] (24) E [X] = 0 (1 F X (x)) dx There are two interesting observations of Theorem 2. Firstly, the theorem applies to any fading distribution. This can also be seen from the proof. Secondly, in order to achieve this asymptotic limit, there is no requirement of optimal rate allocation. This suggests as N becomes large, the gain of optimal rate design diminishes. In a slow fading environment, if one is not allowed to perform coding over different fading states, it is well accepted that the outage performance is a good measure. In INR, the coding is still within one fading state. However, with the use of ARQ feedback, the performance measure switches from outage capacity to ergodic capacity, as the amount of ARQ retransmissions increases to infinity. On the other hand, this

13 13 R 1 + R 2 Primary Transmission ARQ Retransmission Power Power P 2 P 1 R 1 Fig. 1. Illustration of the power allocation for INR with N = 1. R 1 + R 2 and R 1 are the corresponding rates of primary transmission and ARQ retransmission, respectively. result is not surprising in the following two aspects. Firstly, with infinitely many retransmissions allowed, in each fading block the transmission rate is gradually reduced (by NACK from the receiver) until it arrives at the exact rate that the current channel gain g can support: log (1 + gp ). That is, eventually the transmission happens with a rate that perfectly matches to the instantaneous channel realization. Thus all the information nats can theoretically be decoded without any errors, and the average rate is the ergodic capacity. Secondly, allowing ARQ feedback effectively informs the transmitter not only the decoding status, but also a partial CSI. As the number of ARQ retransmissions goes to infinity, the informed CSI becomes perfect eventually. E. Power Allocation for INR with N = 1 In the previous sections, the general idea of RO-HARQ is discussed and the average rate performance is optimized, under the assumption that the transmit power is constant throughout the entire process. Intuitively, allowing power allocation among the primary transmission (step 1) and ARQ retransmissions (step 2,, N + 1) could further improve the average rate. For example, boosting the power of primary transmission will increase the possibility that the primary transmission succeeds. However this comes at the price of decreasing the power of ARQ retransmissions and hence the probability for success, if the average transmit power is kept constant. There is obviously a balance in allocating power among these transmissions, and this idea is pursued in the following. For simplicity our discussion is restricted to INR with N = 1. Extension to general N > 1 is possible, although the optimization problem is more difficult to formulate and solve. Since N = 1, the entire transmission is divided into two sections: primary transmission (with power P 1 ) and ARQ retransmission (with power P 2 ). Figure 1 is a graphical illustration of this process. The retransmission happens only if the primary one fails. Other notations are the same as before. To derive the optimal power allocation that maximizes the average rate, two sub-problems need to be addressed.

14 14 1) Average rate for a given power allocation (P 1, P 2 ): Rate R 1 + R 2 can be reliably decoded if log (1 + gp 1 ) R 1 + R 2. If the primary transmission with power P 1 fails, additional symbols are sent with power P 2, reducing the total transmission rate to R 1. The achievable rate of sending a Gaussian code where the first R 1 /(R 1 + R 2 ) portion has power P 1 and the remaining R 2 /(R 1 + R 2 ) portion has power P 2 is not obvious. However, with a random coding and typical set decoding argument [27], this achievable rate can be shown to be a TDMA-type one: R 1 R 1 + R 2 log (1 + gp 1 ) + R 2 R 1 + R 2 log (1 + gp 2 ). Thus for a given power allocation (P 1, P 2 ), the average rate is given by { R INR,P 1 A = (R 1 + R 2 ) P {log (1 + gp 1 ) R 1 + R 2 } + R 1 P log (1 + gp 1 ) < R 1 + R 2, R 1 log (1 + gp 1 ) + R } 2 log (1 + gp 2 ) R 1. (25) R 1 + R 2 R 1 + R 2 2) Average power constraint: The power allocation (P 1, P 2 ) is a designable parameter. In order to make a fair comparison to the scheme discussed in Section IV-C with a constant power P, it is reasonable to put a constraint on (P 1, P 2 ) such that the average power equals P. The interesting observation of the HARQ scheme is that with unequal power allocation the actual power consumed in the entire transmission is a random variable. The reason is that although the primary transmission with power P 1 happens with probability 1, the ARQ retransmission with power P 2 takes place only if the primary one fails, which is a random event determined by the random fading channel. Denote the actual consumed power as β, then this discrete random variable is P 1, if log (1 + gp 1 ) R 1 + R 2 β = R 1 R 1+R 2 P 1 + R2 R 1+R 2 P 2, otherwise Thus the average power constraint that (P 1, P 2 ) should satisfy is P = E {β} = P 1 P {log (1 + gp 1 ) R 1 + R 2 } ( R1 + P 1 + R ) 2 P 2 P {log (1 + gp 1 ) < R 1 + R 2 }. (27) R 1 + R 2 R 1 + R 2 Notice that in the case of constant power allocation P 1 = P 2 = P, the randomness of β disappears: β = P with probability 1. (26)

15 15 3) Optimal power allocation: Finally, the average rate maximization problem under optimal power allocation is maximize Equation (25) (28) subject to Equation (27). Numerically solving this optimization problem can be made easier by observing that the equality constraint (27) eliminates one variable, and thus the problem becomes a three-variable unconstrained optimization. Optimal average rate of problem (28) is better than that of problem (16): choosing P 1 = P 2 = P makes (25) equal to (15). It is then interesting to ask how big the power allocation gain is. Numerical examples in Section V shows that for the Rayleigh fading channel, this gain is remarkable in the low-snr regime, while it is negligible for medium to high SNR. This is a reasonable result for most of the known power allocation schemes. V. RO-HARQ: FIGURE OF MERIT AND NUMERICAL PERFORMANCE COMPARISON The key idea behind the proposed RO-HARQ scheme is to adapt the transmit signal to channel conditions via the ARQ feedback. Due to the lack of full channel information at the transmitter, such adaptation is not guaranteed to support reliable transmission all the time. Thus the instantaneous successful transmission rate is a random variable, whose distribution is determined by the random channel fading and the rate assignment. As a performance metric, the average rate characterizes the mean value of this random variable. On the other hand, it is arguable that the outage probability (P out ) serves as a worst-case performance measure for the RO-HARQ scheme, as it describes the probability that the transmission fails after the maximum number of ARQ retransmissions are used. Thus, although the analytical discussion of this paper is focused on the average rate maximization, numerical results for different performance measures are reported in this section. To be specific, three performance metrics, average rate [2], [3], [5], [7], outage probability [1], [28], and average rate versus outage probability [3], are used to numerically compare the proposed RO-HARQ with SLT and MLT. Such a comprehensive comparison consolidates the advantages of RO-HARQ over the conventional schemes. Another well-known performance metric for the ARQ system, which is not exploited in this work, is the throughput. It is defined as the ratio of the number of successfully received information nats to the averaged packet length over transmissions [4, Chapter ] [14]. Mathematically, the throughput η is where ξ is an event defined as η = K bp {ξ} E {T } ξ = all K b information nats are successfully transmitted at the end of HARQ, (29)

16 16 and E {T } is the average time for the HARQ process. The throughput defined in this way of several HARQ systems and the design of channel codes for them have been studied in literature, e.g., [14], [18], [23]. It is interesting to point out that these two metrics, throughput and average rate, are in fact related, but they reflect different system performance measures. To gain some insight, let us first rewrite the throughput expression (29) as η = K bp {ξ} E {T } = P { {ξ} } = E T Kb and then similarly rewrite the average rate as P {ξ} E { 1 R(g) ξ } (30) R = E {R(g)} = P {ξ} E {R(g) ξ}. (31) Expressions (30) and (31) suggest that the connection between throughput and average rate is similar to the relationship between 1/E {X} and E {1/X}, and it can be characterized using the Jensen s inequality [27] on function 1/X: η = P {ξ} E { 1 R(g) ξ } P {ξ} E {R(g) ξ} = R. (32) Since 1/X is strictly convex in X, we can conclude that average rate is strictly larger than throughput except when the channel has no fading. The difference between them lies in the practical meaning of the measures. The physical meaning of average rate is that each term in the expectation is an instantaneous transmission rate, which is the actual data rate if the current transmission succeeds. This instantaneous rate cannot be reflected by throughput. Meanwhile, R is the coding rate one can expect if sending many packets with capacity-approaching codes over quasi-static fading channels. The throughput, however, is not directly related to the channel code rate. On the other hand, the calculation of throughput involves E {T }, which has the practical meaning of being the average delay of the HARQ protocol. Thus, throughput and average rate are related but different performance measures. A. Average Rate Numerical optimizations are performed to maximize the average rate of the schemes analyzed in Section III and IV. In the case of slow Rayleigh fading channel with h CN (0, 1), Figure 2 reports the average rate comparison among SLT, MLT, INR and RTD. Ergodic capacity is also shown. It is clear that allowing INR ARQ feedback increases the average rate substantially. For example, even INR with

17 17 Average Rate [bpcu] SLT MLT 2 levels MLT inf levels INR N=1 INR N=2 INR N=4 RTD N=1 RTD N=4 ergodic capacity SNR [db] Fig. 2. Average rate (bpcu) versus average receive SNR (db) of SLT, MLT, INR and RTD in a quasi-static Rayleigh fading channel. Equal power allocation among transmissions is performed. N = 1 outperforms the infinite-layer MLT by half a bit per channel use over a wide range of SNR (15 to 35 db), and is 1.5 bits better than the SLT scheme with optimized rate. Notice that this average rate advantage of INR over MLT does not bring much higher complexity. MLT requires complicated encoding and decoding processes to handle the multiple layers, and this complexity increases with the number of layers. Although ARQ requires feedback and some overhead in the protocol design, typically its complexity is comparable to MLT. The average rate is boosted by another 0.5 bits if N increases to 2. As N further increases, the average rate continues to increase up until the the ergodic capacity. At the same time, RTD type of HARQ is shown to be inefficient in terms of the average rate performance. It is better than SLT 6, but in some configurations is even worse than MLT. This also numerically confirms Theorem 1. For this suboptimality, RTD is not considered in the remaining of the numerical simulations. Figure 3 compares the optimized average rate performance of SLT, MLT and INR in a slow Ricean fading channel. Two different K factors are considered: Figure 3(a) for K = 5, and Figure 3(b) for K = 10. Ergodic capacity is plotted as the performance upper bound. As opposed to the Rayleigh 6 This can be easily proved by looking at equation (20). When n = 1 the component has the same form as SLT.

18 SLT MLT 2 levels INR N=1 INR N=2 INR N=4 ergodic capacity SLT MLT 2 levels INR N=1 INR N=2 INR N=4 ergodic capacity 8 8 Average Rate [bpcu] 6 Average Rate [bpcu] SNR [db] SNR [db] (a) K=5 (b) K=10 Fig. 3. Average rate (bpcu) versus average receive SNR (db) of SLT, MLT and INR in a quasi-static Ricean fading channel with different K factors. Equal power allocation among transmissions is performed. fading case, Ricean fading channel has a high-power line-of-sight (LOS) path and thus is less random. Numerical results show that in the Ricean fading environment, MLT with 2 levels has almost negligible gain over SLT, while INR still performs much better than both SLT and MLT. Combined with its high complexity, this result indicates the inefficiency of MLT in a less random channel environment such as Ricean distribution. However, INR continues to perform very well even with N 1. It seems INR is more robust to the channel fading distributions than MLT. The transmission strategies studied in Section III and IV are for single-antenna systems. Without too much difficulty one can realize that these results can be directly extended to Single-Input Multiple-Output (SIMO) fading channels. Consider a SIMO Rayleigh fading channel with L r receive antennas. It is shown in [29], [30] that the PDF of the total fading power is f Lr (g) = 1 Γ (L r ) glr 1 e g, g 0 where Γ (L r ) = 0 t Lr 1 e t dt is the Gamma function. Figure 4 shows the average rate performance of SLT, MLT, INR and ergodic capacity of a SIMO Rayleigh fading channel with L r = 2 or L r = 4. Several interesting observations can be made from these plot. First, similar to the previous case, MLT with 2 levels has negligible gain over SLT: the performance difference is almost indistinguishable. This again seems to suggest that the gain of MLT with 2 levels is not important in most of the fading distributions, especially considering that MLT is much more complicated than SLT. Secondly, the INR scheme performs

19 SLT MLT 2 levels INR N=1 INR N=2 ergodic capacity 12 SLT MLT 2 levels INR N=1 INR N=2 ergodic capacity 10 Average Rate [bpcu] Average Rate [bpcu] SNR [db] SNR [db] (a) 2 Receive Antennas (b) 4 Receive Antennas Fig. 4. Average rate (bpcu) versus SNR (db) of INR in a quasi-static SIMO Rayleigh fading channel, L r = 2 or L r = 4. extraordinary well: the gap between N = 1 INR with the ergodic capacity is only 0.6 bits and 0.7 bits at the medium to high SNR regime with L r = 2 and L r = 4, respectively. This complies with our previous observation that INR is robust to the channel fading distributions. Numerical results for optimal power allocation of INR with N = 1, and the comparison to equal power allocation, are reported in Figure 5(a) (medium-to-high-snr regime) and Figure 5(b) (low-snr regime) for the Rayleigh fading distribution. The advantage of optimal power allocation is mainly reflected in the low-snr regime. The average rate is tripled when SNR is as low as -25dB. This gain diminishes as SNR increases, and becomes negligible in the medium-to-high-snr regime. B. Outage Probability The outage probability comparison is reported in Figure 6 for both SLT and INR in a Rayleigh fading channel. It should be noted that the outage probability does not apply to MLT, where the requirement that all transmitted data must be decoded is dropped, and thus the concept of outage does not hold anymore. Both SLT and INR are still optimized in terms of the average rate. Thus, the outage event of SLT is { } P log (1 + gp ) < RSLT,Rayleigh where RSLT,Rayleigh is given in (7), and the outage event of INR with N is P {log (1 + gp ) < R 1 } where R 1 is the solution of R 1 in problem (16). The advantage of INR is now more obvious: not only does it increase the average rate, it also decreases the outage probability simultaneously. In fact, with the argument made in Section IV-D, the asymptotic outage probability will be zero, as every transmission will eventually match perfectly with the instantaneous channel condition.

20 constant power allocation optimal power allocation 10 0 constant power allocation optimal power allocation Average rate [bpcu] Average rate [bpcu] SNR [db] SNR [db] (a) Medium-to-high-SNR regime (b) Low-SNR regime Fig. 5. channel. Average rate (bpcu) versus SNR (db) of N = 1 INR with optimal power allocation in a quasi-static Rayleigh fading 10 0 SLT INR N=1 INR N=2 INR N=4 P out SNR [db] Fig. 6. Outage probability versus SNR (db) of INR (N =1, 2, and 4) compared to SLT in a quasi-static Rayleigh fading channel. The two schemes are optimized in terms of the average rate.

21 21 C. Average Rate versus Outage Probability When transmitting over a slow fading channel, the successfully transmitted data rate is a random variable. Its instantaneous value depends on both the instantaneous channel power g and the communication schemes (e.g., SLT, MLT, or HARQ). Intuitively speaking, the average rate somehow describes the mean value of the random performance, while the outage probability characterizes the variance in the sense that it gives the worst-case performance. Thus, to have a comprehensive view and comparison of several schemes, both average rate and outage probability should be jointly considered. Average rate versus outage probability was proposed in [3] as a meaningful association between average rate and outage probability. This metric requires examining the average rate when the channel gain g is known to exceed some threshold g th. This can be viewed as a conditional average rate where the distribution of channel gain F (g) is replaced with the conditional CDF F gth (u) =. P {g u g g th }. Such a conditional average rate examines the average rate with a given outage probability, and thus effectively relates these two metrics. Figure 7 reports the average rate performance where the threshold g th is chosen such that the outage probability is 1% and 30%, respectively. The use of INR still provides remarkable gains with this metric. VI. HOW TO USE THE FEEDBACK CHANNEL: ARQ VERSUS CSI From the information-theoretic point of view, the proposed RO-HARQ is one form of utilizing the feedback link in a wireless communication system. It falls into the general category of quasi-static fading channel with quantized feedback. The capacity (under different definitions, e.g., outage capacity, expected capacity, etc.) of such channel is still unknown in general. Thus it is difficult to determine how well RO- HARQ performs. A reasonable approach would be to compare RO-HARQ with other schemes that utilize the quantized feedback channel to improve performance. One well-known approach is the CSI feedback scheme [6]: the receiver sends an M-bit quantization of the channel state information to the transmitter before the transmission takes place, and the transmitter adjusts its rate according to this imperfect CSIT. To be specific, the set of all possible fading power G = [0, ) is divided into 2 M nonoverlapping subsets G = 2 M i=1 G i, where G i = [d i 1, d i ), d 0 = 0, d 2 M =. If the instantaneous channel power g G i, the receiver sends the index i to the transmitter using the M-bit feedback channel, and the transmitter selects a codeword with rate R i. Due to the imperfect CSI feedback, the transmission could fail if the data rate R i is not supported by the instantaneous channel realization g. It is then natural to ask the following question: since ARQ and CSI feedback are different forms of utilizing feedback in a wireless system, what are the advantages/disadvantages of them, and which one

22 22 15 Average Rate [bpcu] 10 SLT, 1% MLT 2 levels, 1% INR N=1, 1% INR N=2, 1% INR N=4, 1% SLT, 30% MLT 2 levels, 30% INR N=1, 30% INR N=2, 30% INR N=4, 30% SNR [db] Fig. 7. Average rate versus outage probability for SLT, MLT and INR in a quasi-static Rayleigh fading channel. Two thresholds corresponding to 1% and 30% outage probability, respectively, are simulated. is preferred? A few perspectives are discussed in this section. A. System implementation ARQ is a technique in the data link layer, which is implemented in most wireless protocols, e.g., the latest IEEE e standard [31]. Thus exploiting ARQ feedback does not require additional system implementations, other than that the physical layer channel coding should be able to provide incremental redundancy, which has been well studied [15] [20], [26]. CSI feedback, on the other hand, belongs to the physical layer techniques. To exploit CSIT, typically the physical layer needs some additional designs, especially at the transmitter, to adjust the transmission based on the CSI feedback. For example, linear precoder design is needed to rotate the transmit signal in case of multiple transmit antennas [32]. This increases the system complexity and cost. From this perspective, utilizing ARQ seems to be more favorable, since ARQ is already provided by the wireless protocol in the system and no additional closedloop design is needed.

23 23 B. Position of feedback In the RO-HARQ scheme, the ACK/NACK feedbacks are scattered into the entire transmission. There are several hand-shakings between the transmitter and receiver. Each such coordination adds additional overhead. The typical form of CSI feedback, however, is to send the entire CSI bits before the transmission takes place, such that the transmitter adjusts the parameters to match the partially-known channel. From the practical point of view, the latter is more favorable, since all the feedback bits are sent in one shot, which simplifies the protocol, reduces the processing delay, system overhead, and the implementation complexity. C. Amount of feedback To make a fair comparison of the amount of feedback required for INR and CSI feedback, it is assumed that both schemes are optimized for maximum average rates, and the maximum average rates are set to be equal. This is reasonable since the resulting comparison indicates that in order to achieve the same optimal performance, how much feedback is needed for each scheme. The following theorem gives the relationship between N and M under this condition. Theorem 3: For any given fading distribution, if the optimal average rates of (1) INR scheme with maximum N retransmissions, (2) CSI feedback scheme with an M-bit feedback channel are the same, i.e., RN INR = R M CSI, then N + 1 = 2 M. (33) Proof: See Appendix B. From Theorem 3, it seems INR requires more feedback than the CSI feedback scheme: with condition (33) satisfied, M = N if and only if M = N = 1, and N > M strictly holds in all other cases. However, N > M is not necessarily translated to the conclusion that INR requires more feedback. First of all, the actual number of retransmissions associated with INR is a random variable, which is determined by the instantaneous channel realization and the rate assignment. The number N only denotes the maximum number of retransmissions, which is a worst-case constraint. On the other hand, M is a fixed number in the CSI feedback scheme. It is not affected by the channel realization or the rate assignment. Secondly, N retransmissions is not necessarily equivalent to N bits feedback, as the number of physical bits required for N retransmissions is determined by many issues. Wicker [4, Chapter 15.2] discussed several practical issues that might require different number of bits for ACK/NACK. For example, NACK may be set as