Re-Transmission Diversity Multiple Access based on SIC and HARQ-IR

There have been various coordinated multiple access schemes to share a common radio resource block (in the frequency or time domain) in wireless communications, especially for uplink transmissions [1]. As opposed to coordinated multiple access, it is often desirable to consider uncoordinated multiple access to accommodate a number of users of low activity with limited radio resource for packet communications using random access schemes [2]. For example, slotted ALOHA [3] and carrier sense multiple access (CSMA) [4] are widely studied for random access. While random backoff plays a key role in mitigating multiple access interference (MAI), which results in packet collision, for most random access schemes (such as slotted ALOHA and CSMA), multiuser detection and other signal processing approaches can be used to mitigate MAI when packets from multiple users are collided in random access. Similar to IDMA, network-assisted diversity multiple access (NDMA) [5] employs multiuser detection with re-transmissions for diversity combining. In particular, if there are $M$ collided packets at a time, users re-transmit the collided packets $M$ times in total so that a receiver is able to separate them.

While re-transmission diversity¹¹1RTxD is used as the abbreviation of re-transmission diversity to differentiate it from repetition time diversity (RTD). (RTxD) is exploited to separate collided signals in NDMA, it has also been employed in link-layer protocols for single-user transmissions. For example, in hybrid automatic repeat request (HARQ) protocols, RTxD together with channel coding plays a crucial role in providing reliable transmissions over fading channels [6, 7]. HARQ can also be applied to random access [8], where RTxD with incremental redundancy (IR) blocks allows to achieve the channel capacity. It is shown in [9] that the rate optimization can improve the performance of ARQ protocols in terms of outage probability and throughput. In [10], a performance analysis of HARQ protocols is carried out under block-fading, where closed-form expressions are derived for the throughput and outage probability. In [11], closed-form expressions are obtained for the throughputs of various HARQ protocols including those in [9]. HARQ is applied to a multiuser system with multiple orthogonal channels in [12], where resource allocation is considered with HARQ to improve the network outage and fairness.

The notion of RTxD is also used to improve the probability of transmission success in slotted ALOHA [13]. In conjunction with RTxD, signal processing approaches at a receiver such as successive interference cancellation (SIC) can be used to suppress the MAI from collided packets [14, 15].

In this paper, we consider multiple access with both HARQ-IR and SIC that can exploit RTxD and mitigate MAI, respectively. The resulting RTxD multiple access scheme is similar to that in [8] except for SIC. It can also be seen as a generalization of that in [14] with re-transmissions of IR blocks instead of identical coded blocks for better performances. The proposed scheme can not only take advantage of IR to have a smaller number of re-transmissions, but also exploit SIC to effectively mitigate MAI. With the proposed scheme, we find the conditions to guarantee successful decoding of all active signals within a certain number of re-transmissions of IR blocks from active users. While we mainly focus on the application of the proposed scheme to random access in this paper, the proposed scheme can also be employed for coordinated multiple access where conditions for successful decoding of all active signals within a certain number of re-transmissions can be imposed by a scheduler. In this case, we can show that the spectral efficiency can be higher than that of orthogonal multiple access due to SIC.

The main contributions of the paper are as follows: i) We apply HARQ-IR to multiple access with SIC at a receiver to derive an RTxD multiple access scheme, which outperforms other existing random access schemes, and find conditions for successful decoding within a certain number of re-transmissions; ii) From those conditions with some special cases, we further derive two multichannel random access schemes that have sub-channels in the power domain and rate domain.

The rest of the paper is organized as follows. In Section II, we present the system model and the proposed scheme for RTxD multiple access with HARQ-IR and SIC. We analyze the length of frame or required number of re-transmissions for successful decoding and consider two special cases of the proposed scheme in Section III. With multiple power levels, we propose a multichannel random access scheme in Section IV. We also derive another multichannel random access scheme with multiple rates in Section V. Both random access schemes are based on two special cases studied in Section III. We present simulation results in Section VI and conclude the paper with some remarks in Section VII.

Notation: Matrices and vectors are denoted by upper- and lower-case boldface letters, respectively. ${\bf I}$ stands for the identity matrix. ${\mathbb{E}}[\cdot]$ and ${\rm Var}(\cdot)$ denote the statistical expectation and variance, respectively. ${\cal C}{\cal N}({\bf a},{\bf R})$ (${\cal N}({\bf a},{\bf R})$) represents the distribution of circularly symmetric complex Gaussian (CSCG) (resp., real-valued Gaussian) random vectors with mean vector ${\bf a}$ and covariance matrix ${\bf R}$.

In this section, we consider a system consisting of a base station (BS) and multiple users for uplink transmissions by sharing a common channel. Denote by $K$ the number of users. We assume that a frame consists of multiple slots. At each frame, a subgroup of users can be active and send signals to the BS. Each active user employs HARQ-IR. For RTxD, an active user keeps sending IR blocks in every slots until the BS sends a positive acknowledgment (ACK). Note that in the conventional HARQ-IR scheme, each user may have an orthogonal radio resource block so that there is no interference from other users. On the other hand, in the proposed scheme, multiple active users share a common radio resource block for multiple access. Thus, the received signal during time slot $t$ is given by

where $h_{k}$ and ${\bf s}_{k,t}$ represent the channel coefficient and IR block, respectively, from user $k\in{\cal I}$ over time slot $t$ to the BS, and ${\bf n}_{t}\sim{\cal C}{\cal N}(0,{\bf I})$ is the background noise (note that the noise variance is normalized for convenience). In addition, ${\cal I}$ denotes the index set of active users in the current frame. We assume that ${\cal I}$ is invariant within a frame and denote by $M$ the number of active users, i.e., $M=|{\cal I}|$. For convenience, throughout the paper, we assume that ${\bf s}_{k,t}\sim{\cal C}{\cal N}({\bf 0},P_{k}{\bf I})$, i.e., Gaussian codebooks are used for IR blocks, where $P_{k}$ is the transmission power of user $k$ and each IR block, ${\bf s}_{k,t}$, is a codeword. In addition, for convenience, we let $\alpha_{k}=|h_{k}|^{2}$ and assume that $\{\alpha_{k}\}$ is known at the BS.

It is clear that conventional HARQ-IR becomes a special case with $|{\cal I}|=1$. That is, the above multiple access scheme reduces to conventional HARQ-IR if there is only one user at a time, say user $k$. In this case (i.e., conventional HARQ-IR), the minimum number of IR blocks required for successful decoding under the assumption of Gaussian codebooks can be given by [8, 16]

where $R_{k}$ is the (initial) rate of user $k$ and $T$ represents the number of re-transmissions. Note that $T_{\rm CH}$ is a random variable (stopping time). As mentioned earlier, multiple users can transmit signals using HARQ-IR without MAI as in (2) if they are assigned to orthogonal radio resource blocks.

At each time slot, the BS is to decode signals from users. As usual in HARQ protocols, if the BS fails to decode signals, it sends a negative acknowledgment (NACK). If the BS can decode certain users’ signals, it broadcasts ACK with the indices of those users. The users whose signals are decoded by the BS do not send signals in the rest of the current frame, while the other active users keep sending IR blocks. The decoding process at the BS is divided into multiple stages, and in each stage, we assume that the BS succeeds to decode signals from an active user with SIC. As a result, we can have

where $\setminus$ represents the set difference (i.e., $B\setminus A=\{x\in B\,|\,x\notin A\}$), ${\cal I}(n)$ denotes the index set of active users whose signals are not decoded at stage $n$, and $k(n)$ represents the index of the user whose signals are successfully decoded at stage $n$. Clearly, we have ${\cal I}{(0)}={\cal I}$ and the number of stages is $M$ (in this case, at each stage, we assume that only one user’s signal is decoded).

In this section, we study the length of frame, denoted by $T_{\rm F}$, or the required number of re-transmissions of IR blocks from the last active users for successful decoding in the proposed RTxD multiple access scheme. Clearly, $T_{\rm F}$ is a random variable that depends on the number of active users, their channel coefficients, and signal powers.

In this section, based on Corollary 1, we derive a multichannel random access scheme with distributed power allocation which suits to random access under the following assumption.

• A1)

  Each user knows his/her channel state information (CSI), i.e., channel power gain, $\alpha_{k}$.

Note that A1) is valid when the coherence time is sufficiently long (the channel gains remain unchanged within a frame) with time division duplexing (TDD) mode. In this case, each user is able to estimate his/her channel coefficient through downlink training (i.e., a pilot signal transmission from the BS) prior to the beginning of a frame based on the channel reciprocity.

In this section, we consider a variation of the RTxD random access based on HARQ-IR in Section II over block-fading channels that vary from one slot to another. In this case, (1) becomes

where $h_{k,t}$ is the channel coefficient from user $k$ to the BS at time slot $t$. We assume that $h_{k,t}$ is independent. Note that we do not consider Assumption A1) in this section. Thus, instantaneous CSI is not exploited by users.

As in (4), for fast block-fading channels, we can have the initial SINR at stage 0 as follows:

where $\alpha_{k,t}=|h_{k,t}|^{2}$. For convenience, we assume that $P_{k}$ is decided to have the same average receive power at the BS for all $k$ as follows:

where $\gamma_{k}={\mathbb{E}}[\alpha_{k,t}]$. Furthermore, we consider the following assumption:

• A2a)

  We assume that $\alpha_{k,t}P_{k}$ has the same distribution for all $k$, i.e., the $\alpha_{k,t}P_{k}$’s are assumed to be iid.

Then, for a given $T$, the average sum rate becomes

where $\psi_{M}(x)={\mathbb{E}}\left[\log_{2}\left(1+\sum_{k=1}^{M}\alpha_{k,t}x\right)\right]$. If there is an active user of a rate $R_{(1)}\leq T\left(\psi_{M}(U)-\psi_{M-1}(U)\right)$, the average number of re-transmissions for successful decoding is equal to or less than $T$. The signals from this user can be removed using SIC after successful decoding. If there is another user of a rate $R_{(2)}\leq T\left(\psi_{M-1}(U)-\psi_{M-2}(U)\right)$, the signals from this users can be decoded and this user’ average number of re-transmissions for successful decoding is equal to or less than $T$. Based on this, we can consider a random access scheme based on HARQ-IR in the rate domain.

Suppose that there are $L$ rates that are given by

where $\delta>0$ is a design parameter and $\psi_{0}(U)=0$. The average sum rate (after $T$ re-transmissions) is given by

An active user can randomly choose one of the rates in ${\cal R}=\{\eta_{1},\ldots,\eta_{L}\}$ and transmits IR blocks encoded at the selected initial rate. The resulting multichannel random access scheme is referred to as rate-domain multiple access (RDMA).

For example, we may consider Rayleigh fading channels for $h_{k,t}$ as in A2). In this case, from [20, 21], we can show that $\psi_{M}(x)=\frac{1}{\ln 2}\sum_{m=1}^{M}e^{\frac{1}{x}}E_{m}\left(\frac{1}{x}\right)$, $l=1,\ldots,L$, where $E_{m}(x)=\int_{1}^{\infty}e^{-xt}t^{-m}dt$ is the exponential integral function of order $m$. Then, the rates in (41) become

Since $E_{m}(x)>E_{m+1}(x)$, we can claim that $\eta_{l}<\eta_{l+1}$, which implies that the condition in (43) can hold for Rayleigh fading channels.

The condition in (44) is valid under Assumption A2a) based on the weak law of large numbers [22]. In particular, for a sufficiently large $T$ and a small $\delta$, there might be a small $\epsilon$ satisfying (44). Consequently, for independent Rayleigh fading channels, the signals from $M$ active users in RDMA can be decoded within $T$ re-transmissions with a high probability if $M\leq L$ and all the active users choose different initial rates.

Note that as in PDMA, there might be two outage events. Their probabilities (and approximation) are the same as those in (35), (36), and (37). An advantage of RDMA over PDMA is that the transmission power may not be high provided that all the $\gamma_{k}$’s are sufficiently high.

We can consider an asymptotic performance of RDMA for a large $T$ to compare its achievable rate to the achievable rate of orthogonal multiple access with HARQ-IR as follows.

In (46), we see that RDMA can have a higher spectral efficiency than orthogonal multiple access (scheduled one without any interference) if $\psi_{L}(U)\geq\psi_{1}(LU)$. The following result shows that this is true.

In Fig. 2, we show $\psi_{L}(U)$ and $\psi_{1}(LU)$ for different values of $L$ when $U=3$ dB for independent Rayleigh fading channels. We can see that RDMA can provide a higher spectral efficiency than orthogonal multiple access.

We have the following remarks.

• •

  If channels are invariant within a frame, RDMA can exploit the channel reciprocity under A1) to take advantage of known CSI. In this case, as in PDMA, each active user can choose one of the available rates in ${\cal R}$ according to the channel gain.

• •

  PDMA can also be used for block-fading channels where channel gains are varying for each slot. In this case, the transmission power is randomly chosen from a set of pre-determined powers.

In this section, we present simulation results under A2) (i.e., for independent Rayleigh fading channels). For performance comparisons, we consider an RTxD random access scheme without SIC (but HARQ-IR is employed) [8], which has the length of frame in (13), and another random access scheme of re-transmissions of identical coded blocks based on [14], which has the length of frame in (14). For convenience, the former scheme is referred to as multiple access without SIC (MA without SIC) and the latter scheme is referred to as repetition with SIC.

Throughout this section, for PDMA, we assume that $A_{L}$ is decided to have $\Pr(\alpha_{k}<A_{L})=0.1$. In addition, for RDMA, $\delta$ is set to $\delta=\frac{\eta_{1}}{5}$.

Fig. 3 shows the average lengths of frame of PDMA, MA without SIC, and repetition with SIC for different values of $L$ when $\gamma=0$ dB, $K=50$, $p_{a}=0.1$, $R=4$, and $\Lambda(R,\Gamma)=10$ (note that the value of $\Gamma$ is decided for given values of $R$ and $\Lambda(R,\Gamma)$ from (25)). It is shown that PDMA can provide a smaller length of frame than the other schemes and the lengths of frame of all the schemes decrease with $L$ at the cost of the transmission power (as will be shown in Fig. 4 (a)). Since $\Lambda(R,\Gamma)$ is set to 10, it is expected that the average length of frame of PDMA is smaller than 10. If $L$ is large (i.e., $L\geq 10$), we can see that the average length of frame of PDMA can be smaller than 10. However, if $L$ is not large, there might be outage events that result in a longer length of frame than the desired one, 10.

In Fig. 4 (a), the average transmission power is shown for different values of $L$. The transmission power increases with $L$, which can be predicted by (34). As mentioned earlier, the lengths of frame of MA without SIC and repetition with SIC also decrease with $L$ as their transmission powers³³3For simulations of MA without SIC, we assume the same transmission power as that in PDMA. increase with $L$. The probability of power collision decreases with $L$ in Fig. 4 (b) as expected by (37) (we note that (37) is a reasonable upper-bound on ${\mathbb{E}}[P_{{\rm out},2}(M)]$).

Fig. 5 shows the performances of PDMA for different values of $\Lambda(R,\Gamma)$ when $\gamma=0$ dB, $K=50$, $p_{a}=0.1$, $R=4$, and $L=20$. We can see that a large $\Lambda(R,\Gamma)$ is desirable for a smaller length of frame of PDMA. In addition, the transmission power can be low at the cost of a longer transmission time.

To see the impact of $K$ on the performances of PDMA, MA without SIC, and repetition with SIC, different values of $K$ are considered and simulation results are shown in Fig. 6 when $\gamma=0$ dB, $L=20$, $p_{a}=0.1$, $R=4$, and $\Lambda(R,\Gamma)=10$. As $K$ increases, the average lengths of frame of all the schemes increase as there are more active users, while PDMA could provide a lower length of frame than $\Lambda(R,\Gamma)$ up to $K=100$ users. Note that since $\Lambda(R,\Gamma)$ is fixed, as shown in Fig. 6 (b), the average transmission power is invariant with respect to $K$.

We now consider the performance of RDMA. Fig. 7 shows the performances of RDMA and MA without SIC for different values of $L$ when $U=3$ dB, $K=50$, $p_{a}=0.1$, and $T=10$. RDMA performs better than MA without SIC in terms of the average length of frame as shown in Fig. 7 (a). In Fig. 7 (b), it is shown that the average initial rate decreases with $L$, where the theoretical one is obtained from (42).

To see the impact of $T$ on the performances of RDMA and MA without SIC, different values of $T$ are considered and simulation results are shown in Fig. 8 when $U=3$ dB, $K=50$, $p_{a}=0.1$, and $L=10$. The gap between the average lengths of frame of RDMA and MA without SIC increases with $T$ as shown in Fig. 8 (a), which shows that RDMA performs much better than MA without SIC, especially for a large $T$. We also see that the average length of frame of RDMA can be smaller than $T$ for large $T$’s, which demonstrates that RDMA can more likely guarantee that the required number of re-transmissions for successful decoding is less than or equal to $T$ when $T$ is sufficiently large.

Finally, in Fig. 9, we show the normalized spectral efficiencies (i.e., the (average) initial rate per user over average length of frame) of PDMA, RDMA, and MA without SIC for different transmission powers with $L=20$, $\gamma=0$ dB, $K=50$, $p_{a}=0.1$, and $\Lambda(R,\Gamma)=T=10$. We can see that PDMA performs better than RDMA. This performance gain results from the power allocation with known CSI (in (24)) in PDMA. It is interesting to note that this gain vanishes when the transmission power is high. In both PDMA and RDMA, the spectral efficiency increases with the average transmission power. This implies that SIC can effectively mitigate interference from other active users. On the other hand, in MA without SIC, the increase of spectral efficiency becomes saturated when the transmission power increases. This saturation indicates that the spectral efficiency mainly depends on the MAI in MA without SIC and, as a result, the increase of the transmission power does not help improve the spectral efficiency unless some interference cancellation techniques (e.g., SIC) are employed.

In this paper, we studied a multiple access scheme that are based on HARQ-IR and SIC to exploit RTxD and mitigate MAI, respectively. We found conditions that can guarantee a certain number of re-transmissions in the proposed multiple access scheme. Based on the derived conditions and two special cases, we considered two multichannel random access schemes, PDMA and RDMA, that have sub-channels in the power and rate domains, respectively. The analysis results showed that the proposed multiple access scheme can have a shorter length of frame than other existing similar schemes, which implies that the proposed multiple access scheme outperforms other existing ones in terms of the spectral efficiency. We also demonstrated that PDMA and RDMA can be used for coordinated multiple access. In this case, RDMA can have a higher spectral efficiency than orthogonal multiple access.

While we focused on some fundamental properties of the proposed multiple access schemes, we did not consider some possible extensions in this paper. For example, we may consider the case that the BS is equipped with multiple antennas (which may increase the spectral efficiency) and a hybrid approach of PDMA and RDMA in the future as further research topics.