Achievable Diversity Order of HARQ-Aided Downlink NOMA Systems

Non-orthogonal multiple access (NOMA) has been recently identified as a promising candidate for future cellular multiple access [1, 2, 3, 4, 5]. With the paradigms of Internet of Things (IoT), ultra-dense heterogeneous networks (HetNets), etc., many potential NOMA related technologies have been developed for various new scenarios [6, 7, 8]. Unlike the traditional orthogonal multiple access (OMA) techniques, the NOMA technique capitalizes on the superposition coding to accommodate multiple users simultaneously with different power levels yet at a sharing physical resource (frequency/time/code) [9, 10, 11, 12, 13]. Furthermore, the successive interference cancellation (SIC) is performed at the decoding stage to differentiate the desired signal from interfering signals. NOMA not only exploits the multi-user diversity to increase the spectral efficiency, but also seeks to strike a balance between the throughput and user fairness. The notable advantages of NOMA inspire us to integrate it with other potential 5G techniques to fulfill unprecedent challenges posed by new requirements of quality of service (QoS) [14, 15]. In particular, the provision of the ultra-reliability has attracted enormous interest in emerging communication systems. For instance, to accommodate many critical ultra-reliable and low-latency communications (uRLLC) services (e.g., industrial control and traffic safety), the outage probability is expected to be as low as $10^{-9}$ [16]. To this end, reliable communications for NOMA systems can be realized by adopting the retransmission mechanism [17]. The representative retransmission technique is hybrid automatic repeat request (HARQ), which leverages the forward error correction (FEC) at the physical layer and the automatic repeat request (ARQ) at the link layer together[18]. According to whether the retransmitted packets keep unchanged and which combining technique (diversity/code combining) is applied for decoding, HARQ scheme can be further divided into three basic categories, including Type I HARQ, HARQ with chase combining (HARQ-CC) and HARQ with incremental redundancy (HARQ-IR). Specifically, Type I HARQ utilizes the currently received packet for decoding, while HARQ-CC and HARQ-IR employ maximal ratio combining (MRC) and code combining for joint decoding with the erroneously received packets, respectively. To assure the reception reliability of new communication paradigms, it is of practical significance to evaluate the outage performance of different types of HARQ-aided NOMA systems.

The performance of HARQ-assisted NOMA systems has recently received considerable research attention in previous literature. More specifically, the system-level simulations were conducted to demonstrate potential gains of the NOMA scheme over the conventional OMA scheme by considering the link adaptation functionality of HARQ [19]. Moreover, in order to ensure the improvement in the combined signal-to-interference-plus-noise ratio (SINR) in retransmissions, an opportunistic HARQ strategy was further proposed to assist NOMA in [20]. The simulated results disclosed that the opportunistic HARQ combination can bring NOMA gain improvement in cell throughput around 4%$\sim$5%. Moreover, once the near user succeeds to cancel out the NOMA interference from the far user, the interference-cancelled data can be recycled to support the far user through cooperative communications. For instance, D. Roh et al. in [21] assumed that the near user is capable of cooperating with the base station to resend the far user’s signal with HARQ-CC in the presence of the decoding failure at the far user. The proposed cooperative retransmission method improves the outage probability, symbol error rate and throughput of the far user. In addition, the outage performance was further investigated for cooperative Type I HARQ-aided NOMA systems by accounting for the effect of aggregate network interference in [22]. The proposed scheme attains an outage performance gain by 21% compared to the non-cooperative one. In [23], the concept of the error exponent was developed by using large deviations for the equivalent assessment of the outage performance of downlink NOMA-HARQ-IR systems. A lower bound was then derived for the error exponent. Based on this result, power allocation was carried out to guarantee a tolerable outage probability. Unfortunately, the analysis of the outage probability for HARQ-aided NOMA systems is rather involved because of the incorporation of multiple fractional random variables. Besides, most of the prior literature conduct the outage analyses for HARQ-aided NOMA systems based on either simulations or approximations [19, 20, 23]. Particularly for HARQ-CC and HARQ-IR-aided NOMA schemes, the relevant outage probabilities are consequently transformed to distributions of the summation/product of multiple fractional random variables. Nevertheless, there are only several numerical approaches available to tackle these problems. To name a few, the outage performance of two-user HARQ-CC-assisted NOMA systems was approximated by using Gaver-Stehfest procedure in [24]. The analytical results then enabled the investigation of the tradeoff between retransmission times and power allocation coefficient. The similar methodology was afterwards employed to study the impact of time correlation upon the outage probability for two-user partial HARQ-assisted NOMA systems, where HARQ-CC and HARQ-IR are partially implemented in [25]. It was found in both [24] and [25] that full time diversity is achievable for HARQ-assisted NOMA systems by assuming perfect SIC, that is, the interfering signals can be completely eliminated when performing SIC. However, since the decoding failures for the farther user’s message were ignored under the assumption of perfect SIC at the near user in [24, 25], this may result in an overestimated performance of time diversity.

In order to satisfy the stringent requirement of the reliability, the condition of high signal-to-noise ratio (SNR) is usually indispensable for the fulfillment of an exceedingly low target outage probability (e.g., less than $10^{-9}$). This motivates us to investigate the asymptotic characteristics of the outage probability for HARQ-aided NOMA systems in high SNR regime. In particular, the diversity order is a fundamental asymptotic reliability metric to characterize the degree of freedom for a communication system. More specifically, the diversity order describes how the outage probability scales with the transmit power under high SNR. Although it was proved in [26, 27] that full time diversity can be achieved by any type of the three conventional HARQ schemes, this result may not carry over to the HARQ-aided NOMA systems. Hence, a natural question arises here: Can full diversity be achieved by HARQ-aided NOMA systems? To answer this crucial question, it is mandatory to study the diversity order for HARQ-aided NOMA systems.

This paper thoroughly examines the achievable diversity orders for three types of HARQ-aided downlink NOMA systems over Rayleigh fading channels, including Type I HARQ, HARQ-CC and HARQ-IR schemes. First, the diversity order of two-user HARQ-aided NOMA systems is analyzed. Towards this goal, we have to obtain the outage probabilities for various HARQ-aided NOMA systems. Specifically, the outage probability of the Type I HARQ-aided NOMA system can be expressed in closed-form. However, it is nearly impossible to provide compact expressions for outage probabilities of the HARQ-CC and HARQ-IR-aided NOMA systems. This is due to the difficulty of dealing with the summation/product of multiple fractional random variables. Alternatively, the upper and lower bounds of the outage probabilities for the HARQ-CC and HARQ-IR-aided NOMA systems are got by recursively using the integration domain partition approach. Then combining the bounds with squeeze theorem yields the diversity order. Later, we go on to extend our results to the cases with an arbitrary number of users. Besides, the influences of power-efficient transmission strategy and imperfect channel state information (CSI) are investigated in depth. In the end, the analytical results are verified by undertaking Monte Carlo simulations.

The main contributions of this paper are listed as follows.

1. 1.

   The diversity order of HARQ-aided downlink NOMA systems is proved to be a decreasing step function of transmission rate given the ratio between the transmit powers allocated to the two users. Moreover, the analytical results reveal that HARQ-IR-aided NOMA systems have the largest diversity order, followed by HARQ-CC-aided and then Type I HARQ-aided NOMA systems.

2. 2.

   As opposed to the conventional HARQ systems that full time diversity is achievable irrespective of the value of transmission rate, HARQ-aided downlink NOMA systems are capable of attaining full diversity given the maximum allowable transmission rate below a threshold. The threshold of the transmission rate depends on HARQ type and the ratio between transmit powers.

3. 3.

   Likewise, the analytical results can be readily extended to a general case with more than two users. In addition, it is interesting to note that the users’ diversity orders obey a descending order according to their respective average channel gains.

4. 4.

   We then propose an advanced power-efficient transmission strategy, which decreases the outage probability of the farther user. However, the resultant diversity order keeps unchanged as under the simple strategy. Besides, our analysis is applicable to the case of imperfect CSI.

The remaining part of this paper is organized as follows. Section II presents the model of HARQ-aided NOMA systems. The diversity order is derived for various types of HARQ-aided NOMA systems in Section III. Section IV validates the theoretical analysis. Eventually, some conclusions are outlined in the last section. Moreover, Table I provides a list of notations used in the paper.

Denote by ${\bf x}_{i}$ the signal intended for destination user $i$, where $i=1,2$. The signals ${\bf x}_{1}$ and ${\bf x}_{2}$ are drawn from randomly generated Gaussian codebooks, and the transmission rate for ${\bf x}_{i}$ is denoted by $R_{i}$. To multiplex the two users at power-domain, the source employs NOMA to superimpose the signals ${\bf x}_{1}$ and ${\bf x}_{2}$ with different power levels. The average transmit powers allocated to the signals ${\bf x}_{1}$ and ${\bf x}_{2}$ are denoted by ${\mathbb{E}}(|{\bf x}_{i}|^{2})=P_{i}$ for $i=1,2$. To conserve the bandwidth and energy consumed on frequent feedbacks of instantaneous CSI, this paper considers that only the statistical CSI knowledge (i.e., the average channel gains) is known to the source. Meanwhile, the HARQ scheme is adopted to resend the superimposed message under bad channel condition to guarantee the transmission reliability. The maximum number of transmissions for each message is allowed up to $K$ to avoid network congestion. We denote the channel gain between the source and user $i$ in the $k$-th HARQ round by $\alpha_{i,k}$. By assuming independent Rayleigh fading channels, the channel gain $\alpha_{i,k}$ follows an exponential distribution with the probability density function (PDF) given by

where ${\bar{\alpha}_{i}}$ stands for the average channel gain between the source and user $i$, i.e., ${\bar{\alpha}_{i}}=\mathbb{E}({\alpha_{i,k}})$. For simplicity of analysis, we assume that the channel gains are normalized such that the additive Gaussian white noise (AWGN) has unit variance.

The benefit of using NOMA originates from the fact that the difference between fading channels can be exploited to substantially enhance the spectral efficiency. Without loss of generality, we assume that user 1 is closer to the source than user 2. In other words, user 2 has a larger path loss than user 1, i.e., ${\bar{\alpha}_{1}}>{\bar{\alpha}_{2}}$. Following the principle of NOMA, more transmission power (relative to the target transmission rate) will be allocated to the farther user (user 2) to ensure user fairness. As a result, user 1 has a lower SNR than user 2. After receiving the superimposed message, user 1 capitalizes on the SIC to subtract the interfering signal ${\bf x}_{2}$ before decoding its own message ${\bf x}_{1}$. However, user 2 directly decodes its own message ${\bf x}_{2}$ by treating user 1’s message ${\bf x}_{1}$ as interference.

Due to the involvement of multiple transmissions into the HARQ-aided NOMA scheme, the message ${\bf x}_{2}$ might be recovered before ${\bf x}_{1}$ by both users¹¹1By considering the decoding order of SIC, the delivery of the message ${\bf x}_{2}$ is deemed to be a success if and only if both users succeed to decode ${\bf x}_{2}$.. Clearly, it is unnecessary to resend the message ${\bf x}_{2}$ which becomes redundant in the following HARQ rounds. If positive acknowledgement (ACK) messages for ${\bf x}_{2}$ are fed back from both users to the source, ${\bf x}_{2}$ can be eliminated from the superposition coded signal in the succeeding retransmissions. In other words, only ${\bf x}_{1}$ needs to be retransmitted, which consequently yields the reduction of the power consumption. Nonetheless, since ${\bf x}_{2}$ is transparent to both users in the circumstances, the outage probability would not be influenced no matter whether ${\bf x}_{1}$ or the superposition coded signal is subsequently retransmitted. On the other hand, if user 1 successfully decodes its own message ${\bf x}_{1}$ prior to user 2, the natural idea is that only ${\bf x}_{2}$ needs to be resent instead of superposition coded signal. However, it will complicate the outage analysis due to the cumbersome transmission procedure. Thereafter, to simplify the system model as well as ease the analysis, the transmissions of superposition coded signal are therefore assumed in all HARQ rounds, which is also applicable to the scenario with an arbitrary number of NOMA users. In fact, this assumption can be treated as the worst case to some extent. Besides, it is believed that complicating the analysis by straightforwardly taking into account more complex transmission procedure will dilute the paper’s contribution, but it will be briefly discussed in Subsection III-C4.

By applying different encoding and decoding operations at the transceiver, HARQ scheme can be classified into three types, including Type I HARQ, HARQ-CC and HARQ-IR. More specifically, the conventional Type I HARQ recovers the message by relying on the currently received packet, which amounts to using selection combining (SC). The erroneously received packets will be discarded directly without any necessity of saving them. However, it undoubtedly yields the waste of the useful information embedded in these packets. To combat this issue, this paper assumes the HARQ-aided NOMA system is equipped with a dedicated buffer to store the erroneous packets for subsequent decodings. The buffer not only supports the employment of SC for Type I HARQ-aided NOMA system, but also is necessary to store the messages that should be decoded prior to its own message. To further improve the outage performance, HARQ-CC and HARQ-IR adopt more advanced combining techniques at the transceiver. In particular, the techniques of MRC and code combining are applied to HARQ-CC and HARQ-IR, respectively. It is worth mentioning that HARQ-IR surpasses both Type I HARQ and HARQ-CC in terms of the outage probability notwithstanding the extra complexity cost²²2It is noteworthy that the analysis of the computational complexity not only depends on the type of HARQ scheme but also the kinds of FEC code, where the FEC code could be Reed-Solomon code, convolutional code, turbo code, polar code, etc. The computational complexity of HARQ schemes has been reported in [28, 29, 30, 31]. By taking the polar coded HARQ schemes as examples, the decoding operations of Type I HARQ, HARQ-CC and HARQ-IR schemes at the $K$-th transmission require complexities on the order of $O(L\log L)$, $O(L(K+\log L))$ and $O(KL\log(KL))$, respectively, where $O(\cdot)$ refers to the big-O notation and $L$ denotes the length of the subcodeword transmitted in each HARQ round [32].. In addition, the outage performance of HARQ-CC and -IR schemes essentially benefits from the combining techniques rather than the equipment of buffer. Nevertheless, the buffer is an indispensable part of HARQ-CC and -IR.

From the information-theoretical perspective, the accumulated mutual information achieved by user $i$ for decoding ${\bf x}_{2}$ after $K$ HARQ rounds can be expressed as [18, 33]

where the occurrence of the term ${\alpha_{i,k}}{P_{1}}$ in the denominator is due to the fact that user 1’s message is treated as Gaussian noise whilst decoding ${\bf x}_{2}$.

In addition, only if user 1 successfully reconstructs user 2’s message such that ${I_{1\to 2,K}}\geq R_{2}$, user 1 is able to decode its own message ${\bf x}_{1}$ by perfectly eliminating the interfering signal ${{\bf x}_{2}}$ via SIC. Accordingly, the accumulated mutual information achieved by user 1 for decoding its own message after $K$ HARQ rounds is given by

By using the Gaussian codes and typical-set decoding, the outage event occurs at user 1 after $K$ HARQ rounds if user 1 fails to either subtract the interfering signal ${\bf x}_{2}$ or recover its own message. On the basis of (II-C) and (II-C), the outage probability of user 1, $p_{1,K}^{out}$, can be obtained as

Likewise, the outage event happens at user 2 if and only if user 2 fails to decode its own message. Hence, the outage probability of user 2, $p_{2,K}^{out}$, can be written as

Unfortunately, it is generally intractable to derive closed-form expressions for the outage probabilities of HARQ-aided NOMA systems due to the incorporation of multiple fractional random variables. Instead, we turn to examine the asymptotic characteristics of the outage performance. In this paper, we are interested in the diversity order of HARQ-aided NOMA systems. Specifically, the diversity order is an important performance metric to characterize the asymptotic scaling law of the outage probability with respect to the average transmit SNR. More precisely, the diversity order associated with user $i$ is explicitly given by³³3In the numerical analysis, we can define $\tilde{d}_{i}$ as $${\tilde{d}_{i}}=10\frac{{{{\log}_{10}}p_{i,K}^{out}\left({{{[{P_{i}}]}_{{\rm{dB}}}}}\right)-{{\log}_{10}}p_{i,K}^{out}\left({{{[{P_{i}}]}_{{\rm{dB}}}}+{{\left[\delta\right]}_{{\rm{dB}}}}}\right)}}{{{{\left[\delta\right]}_{{\rm{dB}}}}}}$$ and use it to approximate the diversity order $d$, where $\delta$ is a power increment, $p_{i,K}^{out}\left({{{[{P_{i}}]}_{{\rm{dB}}}}}\right)$ denotes the outage probability given ${P_{i}}$ in decibels. This is due to the fact that ${\tilde{d}_{i}}$ commonly has a faster convergence than the definition in (12) as $P_{i}$ approaches to infinity.

From (12), the diversity order quantifies the decreasing slope of the outage curves against the average transmit SNR on a log-log scale. More specifically, the outage probability at high SNR behaves asymptotically as $p_{i,K}^{out}\simeq{\left(\mathcal{C}\cdot P_{i}\right)^{-d_{i}}}$ [34, eq.(3.158)], [35, eq.(1)], where $\mathcal{C}$ is termed as the coding gain. Although the authors in [26, 27] proved that the three conventional HARQ schemes can achieve the same and full diversity order, this result is inapplicable to the HARQ-aided NOMA systems. Therefore, it is mandatory to study the achievable diversity order of the HARQ-aided NOMA systems. Moreover, to simplify the analysis as well as ensure the user fairness, as the transmit powers increase, the ratio between them is assumed to be fixed such that $P_{2}=cP_{1}$.

Recalling that different types of HARQ schemes yield distinct expressions of accumulated mutual information, it is necessary to analyze the diversity order of user 2 for each type of HARQ-aided NOMA scheme individually.

According to (10), it is almost intractable to obtain a closed-form expression for the outage probability of user 1 due to the correlation between ${I_{1\to 1,K}}$ and ${I_{1\to 2,K}}$. Although the outage probability $p_{1,K}^{out}$ can not be easily derived, the associated diversity order can be obtained by using the following lemma.

By applying Lemma 3 to (10), we conclude that the diversity order of user 1 is determined by the diversity orders associated with $\Pr\left\{{{{I_{1\to 1,K}}}<{R_{1}}}\right\}$ and $\Pr\left\{{{I_{1\to 2,K}}<{R_{2}}}\right\}$. Evidently, the asymptotic behavior of $\Pr\left\{{{I_{1\to 2,K}}<{R_{2}}}\right\}$ is the same as that of the outage probability of user 2 because the accumulated mutual information ${I_{1\to 2,K}}$ has the similar form as ${I_{2\to 2,K}}$. Hence, it follows that $\Pr\left\{{{I_{1\to 2,K}}<{R_{2}}}\right\}=\Theta({P_{1}}^{-d_{2}})$. In addition, with (II-C), the probabilities $\Pr\left\{{{{I_{1\to 1,K}}}<{R_{1}}}\right\}$ for Type I HARQ, HARQ-CC and HARQ-IR are expressed as the distributions of the maximum, summation and product of $K$ independent exponential random variables, respectively, which degenerate to the key problems of the conventional HARQ schemes. Fortunately, it has been proved in [26, 27] that full diversity can be achieved by the conventional HARQ systems, that is, the diversity order is equal to $K$. Therefore, Lemma 3 indicates $d_{1}=\min\{d_{2},K\}=d_{2}$ owing to $d_{2}\leq K$. Accordingly, the diversity order of the HARQ-aided NOMA system is restricted by the user with the first decoding order (the one with the worse channel quality), namely user 2.