Performance of Hybrid-ARQ in Block-Fading Channels: A Fixed Outage Probability Analysis

ARQ (automatic repeat request) is an extremely powerful type of feedback-based communication that is extensively used at different layers of the network stack. The basic ARQ strategy adheres to the pattern of transmission followed by feedback of an ACK/NACK to indicate successful/unsuccessful decoding. If simple ARQ or hybrid-ARQ (H-ARQ) with Chase combining (CC) [1] is used, a NACK leads to retransmission of the same packet in the second ARQ round. If H-ARQ with incremental redundancy (IR) is used, the second transmission is not the same as the first and instead contains some “new” information regarding the message (e.g., additional parity bits). After the second round the receiver again attempts to decode, based upon the second ARQ round alone (simple ARQ) or upon both ARQ rounds (H-ARQ, either CC or IR). The transmitter moves on to the next message when the receiver correctly decodes and sends back an ACK, or a maximum number of ARQ rounds (per message) is reached.

ARQ provides an advantage by allowing for early termination once sufficient information has been received. As a result, it is most useful when there is considerably uncertainty in the amount/quality of information received. At the network layer, this might correspond to a setting where the network congestion is unknown to the transmitter. At the physical layer, which is the focus of this paper, this corresponds to a fading channel whose instantaneous quality is unknown to the transmitter.

Although H-ARQ is widely used in contemporary wireless systems such as HSPA [2], WiMax [3] (IEEE 802.16e) and 3GPP LTE [4], the majority of research on this topic has focused on code design, e.g., [5], [6], [7], while relatively little research has focused on performance analysis of H-ARQ [8]. Most relevant to the present work, in [9] Caire and Tuninetti established a relationship between H-ARQ throughput and mutual information in the limit of infinite block length. For multiple antenna systems, the diversity-multiplexing-delay tradeoff of H-ARQ was studied by El Gamal et al. [10], and the coding scheme achieving the optimal tradeoff was introduced; Chuang et al. [11] considered the optimal SNR exponent in the block-fading MIMO (multiple-input multiple-output) H-ARQ channel with discrete input signal constellation satisfying a short-term power constraint. H-ARQ has also been recently studied in quasi-static channels (i.e., the channel is fixed over all H-ARQ rounds) [12], [13] and shown to bring benefits to secrecy [14].

In this paper we build upon the results of [9] and perform a mutual information-based analysis of H-ARQ in block-fading channels. We consider a scenario where the fading is too fast to allow instantaneous channel quality feedback to the transmitter, and thus the transmitter only has knowledge of the channel statistics, but nonetheless each transmission experiences only a limited degree of channel selectivity. In this setting, rate adaptation can only be performed based on channel statistics and achieving a reasonable error/outage probability generally requires a conservative choice of rate if H-ARQ is not used. On the other hand, H-ARQ allows for implicit rate adaptation to the instantaneous channel quality because the receiver terminates transmission once the channel conditions experienced by a codeword are good enough to allow for decoding.

We analyze the long-term average transmitted rate achieved with H-ARQ, assuming that there is a maximum number of H-ARQ rounds and that a target outage probability at H-ARQ termination cannot be exceeded. We compare this rate to that achieved without H-ARQ in the same setting as well as to the ergodic capacity, which is the achievable rate in the idealized setting where instantaneous channel information is available to the transmitter. The main findings of the paper are that (a) H-ARQ generally provides a significant advantage over systems that do not use H-ARQ but have an equivalent level of channel selectivity, (b) the H-ARQ rate is reasonably close to the ergodic capacity in many practical settings, and (c) the rate with H-ARQ is much less sensitive to the desired outage probability than an equivalent system that does not use H-ARQ.

The present work differs from prior literature in a number of important aspects. One key distinction is that we consider systems in which the rate is adapted to the average SNR such that a constant target outage probability is maintained at all SNR’s, whereas most prior work has considered either fixed rate (and thus decreasing outage) [11] or increasing rate and decreasing outage as in the diversity-multiplexing tradeoff framework [10][15]. The fixed outage paradigm is consistent with contemporary wireless systems where an outage level near 1% is typical (see [16] for discussion), and certain conclusions depend heavily on the outage assumption. With respect to [9], note that the focus of [9] is on multi-user issues, e.g., whether or not a system becomes interference-limited at high SNR in the regime of very large delay, whereas we consider single-user systems and generally focus on performance with short delay constraints (i.e., maximum number of H-ARQ rounds). In addition, we use the attempted transmission rate, rather than the successful rate (which is used in [9]), as our performance metric. This is motivated by applications such as Voice-over-IP (VoIP), where a packet is dropped (and never retransmitted) if it cannot be decoded after the maximum number of H-ARQ rounds and quality of service is maintained by achieving the target (post-H-ARQ) outage probability. On the other hand, reliable data communication requires the use of higher-layer retransmissions whenever H-ARQ outages occur; in such a setting, the relevant metric is the successful rate, which is the product of the attempted transmission rate and the success probability (i.e., one minus the post-H-ARQ outage probability). If the post-H-ARQ outage is fixed to some target value (e.g., $1\%$), then studying the attempted rate is effectively equivalent to studying the successful rate.¹¹1If the post-H-ARQ outage probability can be optimized, then a careful balancing between the attempted rate and higher-layer retransmissions should be conducted in order to maximize the successful rate. Although this is beyond the scope of the present paper, note that some results in this direction can be found in [17] [18].

We consider a block-fading channel where the channel remains constant over a block but varies independently from one block to another. The $t$-th received symbol in the i-th block is given by:

where the index $i=1,2,\cdots$ indicates the block number, $t=1,2,\cdots,T$ indexes channel uses within a block, SNR is the average received SNR, $h_{i}$ is the fading channel coefficient in the $i$-th block, and $x_{t,i},y_{t,i},$ and $z_{t,i}$ are the transmitted symbol, received symbol, and additive noise, respectively. It is assumed that $h_{k}$ is complex Gaussian (circularly symmetric) with unit variance and zero mean, and that $h_{1},h_{2},\ldots$ are i.i.d.. The noise $z_{t,i}$ has the same distribution as $h_{k}$ and is independent across channel uses and blocks. The transmitted symbol $x_{t,i}$ is constrained to have unit average power; we consider Gaussian inputs, and thus $x_{t,i}$ has the same distribution as the fading and the noise. Although we focus only on Rayleigh fading and single antenna systems, our basic insights can be extended to incorporate other fading distributions and MIMO as discussed in Section IV (Remark 1).

We consider the setting where the receiver has perfect channel state information (CSI), while the transmitter is aware of the channel distribution but does not know the instantaneous channel quality. This models a system in which the fading is too fast to allow for feedback of the instantaneous channel conditions from the receiver back to the transmitter, i.e., the channel coherence time is not much larger than the delay in the feedback loop. In cellular systems this is the case for moderate-to-high velocity users. This setting is often referred to as open-loop because of the lack of instantaneous channel tracking at the transmitter, although other forms of feedback, such as H-ARQ, are permitted. The relevant performance metrics, notably what we refer to as outage probability and fixed outage transmitted rate, are specified at the beginning of the relevant sections.

If H-ARQ is not used, we assume each codeword spans $L$ fading blocks; $L$ is therefore the channel selectivity experienced by each codeword. When H-ARQ is used, we make the following assumptions:

• •

  The channel is constant within each H-ARQ round ($T$ symbols), but is independent across H-ARQ rounds.²²2An intuitive but somewhat misleading extension of the quasi-static fading model to the H-ARQ setting is to assume that the channel is constant for the duration of the H-ARQ rounds corresponding to a particular message/codeword, but is drawn independently across different messages. Because more H-ARQ rounds are needed to decode when the channel quality is poor, such a model actually changes the underlying fading distribution by increasing the probability of poor states and reducing the probability of good channel states. In this light, it is more accurate model the channel across H-ARQ rounds according to a stationary and ergodic random process with a high degree of correlation.

• •

  A maximum of $M$ H-ARQ rounds are allowed. An outage is declared if decoding is not possible after $M$ rounds, and this outage probability can be no larger than the constraint $\epsilon$.

Because the channel is assumed to be independent across H-ARQ rounds, $M$ is the maximum amount of channel selectivity experienced by a codeword. When comparing H-ARQ and no H-ARQ, we set $L=M$ such that maximum selectivity is equalized.

It is worth noting that these assumptions on the channel variation are quite reasonable for the fast-fading/open-loop scenarios. Transmission slots in modern systems are typically around one millisecond, during which the channel is roughly constant even for fast fading. ³³3Frequency-domain channel variation within each H-ARQ round is briefly discussed in Section IV-B. An H-ARQ round generally corresponds to a single transmission slot, but subsequent ARQ rounds are separated in time by at least a few slots to allow for decoding and ACK/NACK feedback; thus the assumption of independent channels across H-ARQ rounds is reasonable. Moreover, a constraint on the number of H-ARQ rounds limits complexity (the decoder must retain information received in prior H-ARQ rounds in memory) and delay.

Throughout the paper we use the notation $F_{\epsilon}^{-1}(X)$ to denote the solution $y$ to the equation $\mathbb{P}[X\leq y]=\epsilon$, where $X$ is a random variable; this quantity is well defined wherever it is used.

We begin by studying the baseline scenario where H-ARQ is not used and every codeword spans $L$ fading blocks. In this setting the outage probability is the probability that mutual information received over the $L$ fading blocks is smaller than the transmitted rate $R$ [19, eq (5.83)]:

where $h_{i}$ is the channel in the $i$-th fading block. The outage probability reasonably approximates the decoding error probability for a system with strong coding [20] [21], and the achievability of this error probability has been rigorously shown in the limit of infinite block length ($T\rightarrow\infty$) [22] [23].

Because the outage probability is a non-decreasing function of $R$, by setting the outage probability to $\epsilon$ and solving for $R$ we get the following straightforward definition of $\epsilon$-outage capacity [24]:

Using notation introduced earlier, the $\epsilon$-outage capacity can be rewritten as

For $L=1$, $P_{\textrm{out}}(R,\mbox{\scriptsize\sf SNR})$ can be written in closed form and inverted to yield $C_{\epsilon}^{1}(\mbox{\scriptsize\sf SNR})=\log_{2}\left(1+\log_{e}\left(\frac{1}{1-\epsilon}\right)\mbox{\scriptsize\sf SNR}\right)$ [19]. For $L>1$ the outage probability cannot be written in closed form nor inverted, and therefore $C_{\epsilon}^{L}(\mbox{\scriptsize\sf SNR})$ must be numerically computed. There are, however, two useful approximations to $\epsilon$-outage capacity. The first one is the high-SNR affine approximation [25], which adds a constant rate offset term to the standard multiplexing gain characterization.

In terms of standard high SNR notation where $C(\mbox{\scriptsize\sf SNR})=\mathcal{S}_{\infty}(\log_{2}\mbox{\scriptsize\sf SNR}-\mathcal{L}_{\infty})+o(1)$ [25][27], the multiplexing gain $\mathcal{S}_{\infty}=1$ and the rate offset $\mathcal{L}_{\infty}=-F_{\epsilon}^{-1}\left(\frac{1}{L}\sum_{i=1}^{L}\log_{2}\left(|h_{i}|^{2}\right)\right)$. The rate offset is the difference between the $\epsilon$-outage capacity and the capacity of an AWGN channel with signal-to-noise ratio SNR. Although a closed form expression for $\mathcal{L}_{\infty}$ cannot be found for $L>1$, from [28],

where $G_{p,q}^{m,n}\left(x|_{b_{1},\ldots,b_{q}}^{a_{1},\ldots,a_{p}}\right)$ is the Meijer G-function [29, eq. (9.301)]. Based on (6), $\mathcal{L}_{\infty}$ therefore is the solution to $2^{-\mathcal{L}_{\infty}L}G_{1,L+1}^{L,1}\left(2^{-\mathcal{L}_{\infty}L}|_{0,0,\ldots,0,-1}^{0}\right)=\epsilon$. The rate offset $\mathcal{L}_{\infty}$ is plotted versus $L$ in Fig. 1 for $\epsilon=0.01$. As $L\rightarrow\infty$ the offset converges to $-{\mathbb{E}}[\log_{2}\left(|h|^{2}\right)]\approx 0.83$, the offset of the ergodic Rayleigh channel [27].

While the affine approximation is accurate at high SNR’s, motivated by the Central Limit Theorem (CLT), an approximation that is more accurate for moderate and low SNR’s is reached by approximating random variable $\frac{1}{L}\sum_{i=1}^{L}\log_{2}(1+\mbox{\scriptsize\sf SNR}|h_{i}|^{2})$ by a Gaussian random variable with the same mean and variance [30][31]. The mean $\mu$ and variance $\sigma^{2}$ of $\log_{2}(1+\mbox{\scriptsize\sf SNR}|h|^{2})$ are given by [32][33]:

where $E_{1}(x)=\int_{1}^{\infty}t^{-1}e^{-xt}dt$, and at high SNR the standard deviation $\sigma(\mbox{\scriptsize\sf SNR})$ converges to $\frac{\pi\log_{2}e}{\sqrt{6}}$ [33]. The mutual information is thus approximated by a $\mathcal{N}(\mu(\mbox{\scriptsize\sf SNR}),\frac{\sigma^{2}(\mbox{\scriptsize\sf SNR})}{L})$, and therefore

where $Q(\cdot)$ is the tail probability of a unit variance normal. Setting this quantity to $\epsilon$ and then solving for $R$ yields an $\epsilon$-outage capacity approximation [30, eq. (26)]:

The accuracy of this approximation depends on how accurately the CDF of a Gaussian matches the CDF (i.e., outage probability) of random variable $\frac{1}{L}\sum_{i=1}^{L}\log_{2}(1+\mbox{\scriptsize\sf SNR}|h_{i}|^{2})$. In Fig. 2 both CDF’s are plotted for $L=2$ and $L=10$, and $\mbox{\scriptsize\sf SNR}=0$, $10$, and $20$ dB. As expected by the CLT, as $L$ increases the approximation becomes more accurate. Furthermore, the match is less accurate for very small values of $\epsilon$ because the tails of the Gaussian and the actual random variable do not precisely match. Finally, note that the match is not as accurate at low SNR’s: this is because the mutual information random variable has a density close to a chi-square in this regime, and is thus not well approximated by a Gaussian. Although not accurate in all regimes, numerical results confirm that the Gaussian approximation is reasonably accurate for the range of interest for parameters (e.g., $0.01\leq\epsilon\leq 0.2$ and $0\leq\mbox{\scriptsize\sf SNR}\leq 20$ dB). More importantly, this approximation yields important insights.

In Fig. 3 the true $\epsilon$-outage capacity $C_{\epsilon}^{L}(\mbox{\scriptsize\sf SNR})$ and the affine and Gaussian approximations are plotted versus SNR for $\epsilon=0.01$ and $L=3,10$. The Gaussian approximation is reasonably accurate at moderate SNR’s, and is more accurate for larger values of $L$. On the other hand, the affine approximation, which provides a correct high SNR offset, is asymptotically tight at high SNR.

We now move on to the analysis of hybrid-ARQ, which will be shown to provide a significant performance advantage relative to the baseline of non-H-ARQ performance. H-ARQ is clearly a variable-length code, in which case the average transmission rate must be suitably defined. If each message contains $b$ information bits and each ARQ round corresponds to $T$ channel symbols, then the initial transmission rate is $R_{\textrm{init}}\triangleq\frac{b}{T}$ bits/symbol. If random variable $X_{i}$ denotes the number of H-ARQ rounds used for the $i$-th message, then a total of $\sum_{i=1}^{N}X_{i}$ H-ARQ rounds are used and the average transmission rate (in bits/symbol or bps/Hz) across those $N$ messages is:

We are interested in the long-term average transmission rate, i.e., the case where $N\rightarrow\infty$. By the law of large numbers (note that the $X_{i}$’s are i.i.d. in our model), $\frac{1}{N}\sum_{i=1}^{N}X_{i}\rightarrow{\mathbb{E}}[X]$ and thus the rate converges to

Here $X$ is the random variable representing the number of H-ARQ rounds per message; this random variable is determined by the specifics of the H-ARQ protocol.

In the remainder of the paper we focus on incremental redundancy (IR) H-ARQ because it is the most powerful type of H-ARQ, although we compare IR to Chase combining in Section IV-E. In [9] it is shown that mutual information is accumulated over H-ARQ rounds when IR is used, and that decoding is possible once the accumulated mutual information is larger than the number of information bits in the message. Therefore, the number of H-ARQ rounds $X$ is the smallest number $m$ such that:

The number of rounds is upper bounded by $M$, and an outage occurs whenever the mutual information after $M$ rounds is smaller than $R_{\textrm{init}}$:

This is the same as the expression for outage probability of $M$-order diversity without H-ARQ in (2), except that mutual information is summed rather than averaged over the $M$ rounds. This difference is a consequence of the fact that $R_{\textrm{init}}$ is defined for transmission over one round rather than all $M$ rounds; dividing by ${\mathbb{E}}[X]$ in (17) to obtain the average transmitted rate makes the expressions consistent. Due to this relationship, if the initial rate is set as $R_{\textrm{init}}=M\cdot C_{\epsilon}^{M}$, where $C_{\epsilon}^{M}$ is the $\epsilon$-outage capacity for $M$-order diversity without H-ARQ, then the outage at H-ARQ termination is $\epsilon$.

In order to simplify expressions, it is useful to define $A_{k}(R_{\textrm{init}})$ as the probability that the accumulated mutual information after $k$ rounds is smaller than $R_{\textrm{init}}$:

The expected number of H-ARQ rounds per message is therefore given by:

The long-term average transmitted rate, which is denoted as $C_{\epsilon}^{\textrm{IR},M}$, is defined by (17). With initial rate $R_{\textrm{init}}=M\cdot C_{\epsilon}^{M}$ we have:⁴⁴4All quantities in this expression except $M$ are actually functions of SNR. For the sake of compactness, however, dependence upon SNR is suppressed in this and subsequent expressions, except where explicit notation is necessary.

Note that $C_{\epsilon}^{\textrm{IR},M}$ is the attempted long-term average transmission rate, as discussed in Section I. For the sake of brevity this quantity is referred to as the H-ARQ rate; this is not to be confused with the initial rate $R_{\textrm{init}}$. Similarly, we refer to $\epsilon$-outage capacity $C_{\epsilon}^{M}$ as the non-H-ARQ rate in the rest of the paper.

Because ${\mathbb{E}}[X]\leq M$, the H-ARQ rate is at least as large as the non-H-ARQ rate, i.e., $C_{\epsilon}^{\textrm{IR},M}\geq C_{\epsilon}^{M}$, and the advantage with respect to the non-H-ARQ benchmark is precisely the multiplicative factor $\frac{M}{{\mathbb{E}}[X]}$. This difference is explained as follows. Because $R_{\textrm{init}}=M\cdot C_{\epsilon}^{M}$, each message/packet contains $C_{\epsilon}^{M}MT$ information bits regardless of whether H-ARQ is used. Without H-ARQ these bits are always transmitted over $MT$ symbols, whereas with H-ARQ an average of only ${\mathbb{E}}[X]T$ symbols are required.

In Fig. 5 the average rates with ($C_{\epsilon}^{\textrm{IR},M}$) and without H-ARQ ($C_{\epsilon}^{M}$) are plotted versus SNR for $\epsilon=0.01$ and $M=1,2$ and $6$ ($M=1$ does not allow for H-ARQ in our model). Ergodic capacity is also plotted as a reference. Based on the figure, we immediately notice:

• •

  H-ARQ with 6 rounds outperforms H-ARQ with 2 rounds.

• •

  H-ARQ provides a significant advantage relative to non-H-ARQ for the same value of $M$ for a wide range of SNR’s, but this advantage vanishes at high SNR.

Increasing rate with $M$ is to be expected, because larger $M$ corresponds to more time diversity and more early termination opportunities. The behavior with respect to SNR is perhaps less intuitive. The remainder of this section is devoted to quantifying and explaining the behavior seen in Fig. 5. We begin by extending the Gaussian approximation to H-ARQ, then examine performance scaling with respect to $M$, SNR, and $\epsilon$, and finally compare IR to Chase Combining.

In this paper we have studied the performance of hybrid-ARQ in the context of an open-loop/fast-fading system in which the transmission rate is adjusted as a function of the average SNR such that a target outage probability is not exceeded. The general findings are that H-ARQ provides a significant rate advantage relative to a system not using H-ARQ at reasonable SNR levels, and that H-ARQ provides a rate quite close to the ergodic capacity even when the channel selectivity is limited.

There appear to be some potentially interesting extensions of this work. Contemporary cellular systems utilize simple ARQ on top of H-ARQ, and it is not fully understood how to balance these reliability mechanisms; some results in this direction are presented in [17]. Although we have assumed error-free ACK/NACK feedback, such errors can be quite important (c.f., [35]) and merit further consideration. Finally, while we have considered only the mutual information of Gaussian inputs, it is of interest to extend the results to discrete constellations and possibly compare to the performance of actual codes.

The authors gratefully acknowledge Prof. Robert Heath for suggesting use of the Gaussian approximation, and Prof. Gerhard Wunder for suggesting use of Chebyshev’s inequality in Section III-A.