Spectral-Efficiency and Energy-Efficiency of Variable-Length XP-HARQ

As illustrated in Fig. 1, in the initial HARQ round, the message ${\rm m}_{1}$ consisting of ${b_{1}}$ information bits is encoded into a codeword ${{\bf x}_{1}}$. The codeword ${{\bf x}_{1}}$ contains ${N_{1}}$ symbols drawn from a constellation set $\mathcal{X}$, i.e., ${{\bf x}_{1}}={\Phi}\left[{\rm m}_{1}\right]\in{\mathcal{X}^{{N_{1}}}}$, where $\Phi[\cdot]$ refers to the encoding operation. If the receiver succeeds to reconstruct the message, a new cycle of HARQ will be triggered by informing the transmitter with a positive acknowledgement (ACK) message. Otherwise, if the receiver fails to decode the message, a negative acknowledgement (NACK) message will be fed back to the transmitter. In order to substantially reduce the transmission latency and boost the spectral efficiency, the transmitter will combine the failed message ${\rm m}_{1}$ with a new message ${\rm m}_{2}$ containing $b_{2}$ information bits to construct a longer message ${{\rm m}}_{[2]}=[{\rm m}_{1},{\rm m}_{2}]$ in the subsequent HARQ round. The message ${{\rm m}}_{[2]}$ is then encoded as a codeword ${\bf{x}}_{2}$ composing of ${N_{2}}$ symbols, i.e., ${{\bf{x}}_{2}}=\Phi\left[{{{\rm m}_{[2]}}}\right]\in{{\mathcal{X}}^{{N_{2}}}}$. At the receiver, the observations of ${\bf{x}}_{1}$ and ${\bf{x}}_{2}$ will be concatenated for joint decoding. If the decoding is successful, the receiver will send back an ACK message and a new HARQ cycle will be immediately initiated. Otherwise, a retransmission is requested by feeding back an NACK message. In the next HARQ round, all the failed messages will be combined with some new information bits for retransmission. The cycle of HARQ will be terminated once the maximum number of transmissions, i.e., $K$, is reached or all the information bits are successfully decoded. Different from [29] that the fixed length of the codeword in each HARQ round is assumed, we consider a more general case that the length of codewords could vary with HARQ rounds.

In this paper, we assume block Rayleigh fading channels. The received signal ${{\bf y}_{k}}$ in the $k$-th HARQ round is given by

where ${P_{k}}$ is the average transmit power in the $k$-th round, ${{\bf z}_{k}}$ is the complex additive Gaussian white noise (AWGN) with zero mean and variance of ${\sigma^{2}}$, ${h_{k}}$ denotes the Rayleigh channel coefficient of the $k$-th HARQ round with normalized average power, i.e., ${\mathbb{E}}\{{{{|{{h_{k}}}|}^{2}}}\}=1$, where ${\mathop{\mathbb{E}}\nolimits}\{\cdot\}$ stands for the expectation operation. According to (1), the received signal-to-noise ratio (SNR) in the $k$-th HARQ round is written as

On the basis of the above transmission protocol and the channel model, the spectral efficiency and the energy efficiency of VL-XP-HARQ are thoroughly investigated in this paper.

In this section, we investigate the relationship between SE and ergodic capacity (EC). According to the protocol of VL-XP-HARQ and from the information-theoretical perspective, the receiver successfully decodes the message after $\mathcal{K}$ HARQ rounds if and only if

To ease analysis, we define $x_{n}=\gamma_{n}$ for $l\in[N_{n-1}+1,\sum_{l=1}^{n}N_{l}]$. Thus, (9) can be rewritten as

The required number of HARQ rounds in one cycle, i.e., $\mathcal{K}$, is a random variable whose probability mass function (pmf) is given by

If we assume that all the messages can be almost surely successfully recovered within $K$ transmissions, i.e., $p_{out,K}\to 0$, taking the expectation of (10a) with regard to $x_{l}$ and $\mathcal{K}$ leads to

For simplicity, we assume equal power allocation scheme for VL-XP-HARQ, i.e., $P_{1}=\cdots=P_{K}$. Moreover, by ignoring the dependence between $x_{l}$ and $\mathcal{K}$, we have

where $\bar{C}={\mathbb{E}}\left\{{{{\log}_{2}}\left({1+{x_{l}}}\right)}\right\}$ denotes the ergodic capacity (EC). By noticing that ${\mathbb{E}}\left\{{\sum\nolimits_{k=1}^{\mathcal{K}}{{N_{k}}}}\right\}=\mathbb{E}\{T\}$ represents the average number of symbols within $\mathcal{K}$ HARQ rounds and ${\mathbb{E}}\left\{b_{\mathcal{K}}^{\Sigma}\right\}=\mathbb{E}\{\mathcal{R}\}+p_{out,K}b_{K}^{\Sigma}\to\mathbb{E}\{\mathcal{R}\}$ stands for the average number of delivered information bits, (13) can be rewritten as

where $\mathbb{E}\left\{{{N_{\cal K}}}\right\}$ and $\mathbb{E}\left\{{{b_{\cal K}}}\right\}$ refer to the average numbers of symbols and information bits in the last HARQ round, respectively. On the basis of (14) along with (4), we arrive at

It is clear from (15) that the SE is less than the EC. In order to achieve the EC, i.e., $\mathcal{T}_{K}=\bar{C}$, according to the squeeze theorem, (15) implies that

Moreover, since (16) is based on the assumption of $p_{out,K}\to 0$, such a lossless HARQ requires $K\to\infty$ [31]. Hence, in order to achieve the EC, as $K\to\infty$, one has

By noticing that ${\mathbb{E}\left\{{{N_{\mathcal{K}}}}\right\}}$ are bounded with the increase of $K$, i.e., $\min\{{N_{k}:k\in[1,K]}\}\leq{\mathbb{E}\left\{{{N_{\mathcal{K}}}}\right\}}=\sum\nolimits_{k=1}^{K-1}{{N_{k}}\left({{p_{out,k-1}}-{p_{out,k}}}\right)}+{N_{K}}{p_{out,K-1}}\leq\max\{N_{k}:k\in[1,K]\}$, it follows from (17) that ${\mathbb{E}\left\{T\right\}}$ should approach to infinity as $K$ increases, i.e., ${\mathbb{E}\left\{T\right\}}\to\infty$. To ensure this condition, we can set $\inf\nolimits_{k\leq K-1}\{p_{out,k}\}>0$ as $K\to\infty$ because of ${{\mathop{\mathbb{E}}\nolimits}\{T\}}>\inf\nolimits_{k\leq K-1}\{p_{out,k}\}\sum\nolimits_{k=1}^{K}{N_{k}}$, where $\inf$ corresponds to the infimum. Similarly, ${\mathbb{E}\left\{{{b_{\mathcal{K}}}}\right\}}$ is also bounded as $\min\{b_{k}:k\in[1,K]\}\leq{\mathbb{E}\left\{{{b_{\mathcal{K}}}}\right\}}\leq\max\{b_{k}:k\in[1,K]\}$, which indicates $\mathbb{E}\{\mathcal{R}\}\to\infty$ as $K$ increases by capitalizing on (17). To guarantee this condition, we can also let $\inf\nolimits_{k\leq K-1}\{p_{out,k}\}>0$, which results in $\mathbb{E}\{\mathcal{R}\}\to\infty$ according to (III).

In summary, in order to attain the ergodic capacity, the outage probability of VL-XP-HARQ can be set as $\sup_{K>0}\inf\nolimits_{k\leq K-1}\{p_{out,k}\}>0$ and $p_{out,K}\to 0$ as $K$ tends to infinity, where $\sup$ stands for the supremum. To meet such conditions, we reach the conclusion that $b_{K}^{\Sigma}\to\infty$ as $K$ increases, while $b_{1},\cdots,b_{K}$ are bounded.

In the last subsection, it is proved that the ergodic capacity is achievable by VL-XP-HARQ. Towards this end, the number of new information bits introduced into each HARQ round, i.e., ${b_{1}},\cdots,{b_{K}}$, should be well designed to maximize the SE while maintaining the outage constraint, i.e., ${{p_{out,K}}\leq\varepsilon}$, where $\varepsilon$ denotes the maximum allowable outage probability. The maximization of the SE is therefore formulated as

where ${\bf b}=(b_{1},\cdots,b_{K})$. To solve the problem of (18), the closed-form expression of the outage probability ${{p_{out,k}}}$ needs to be obtained. However, it is hardly possible to derive a general expression for ${{p_{out,k}}}$ except for the cases of $k\leq 2$, as given below.

Although the outage probability $p_{out,k}$ in the cases of $k=1,2$ can be derived in closed-form, the complex form of the exact outage expression still precludes the optimization of the problem (18). To address this issue, the asymptotic expression of the outage probability in the high SNR regime, i.e., ${P_{1}/\sigma^{2}},\cdots,{P_{K}/\sigma^{2}}\to\infty$, is obtained to ease the optimization of (18), as shown in the following theorem.

In the following, the simple form of the asymptotic outage probability in Theorem 2 is used to replace the exact outage probability $p_{out,k}$ in (18) to reduce the computational complexity of the optimization. It is worth mentioning that the optimal solution to this relax problem will approach to the true optimal solution at high SNR. This is due to the fact that the asymptotic outage probability becomes more accurate as the SNR increases.

By noticing that problem (18) is a fractional programming problem, the Dinkelbach’s transform can be adopted [33]. Thus the optimization problem (18) can be converted into

where $\tilde{\mathcal{R}}({\bf{b}})=\sum\nolimits_{k=1}^{K}{b_{k}}({\tilde{p}_{out,k-1}}({\bf{b}})-{\tilde{p}_{out,K}}({\bf{b}}))$, $\tilde{T}\left({\bf{b}}\right)=\sum\nolimits_{k=1}^{K}{{N_{k}}{\tilde{p}_{out,k-1}}\left({\bf{b}}\right)}$, ${\tilde{p}_{out,k}}(\mathbf{b})$ corresponds to the asymptotic outage probability of VL-XP-HARQ with the numbers of newly introduced information bits set as $\bf b$ and ${\tilde{p}_{out,k}}(\mathbf{b})$ can be evaluated through Theorem 2. Moreover, $\alpha$ is the introduced auxiliary variable, which can be iteratively updated as

where $n$ the iteration index and we denote by ${\bf b}^{(0)}$ the initial value of $\bf b$ for the iteration algorithm. Unfortunately, the problem of (34) is still non-convex. To solve this problem, the alternating optimization [34] is invoked. In particular, the numbers of information bits for different HARQ rounds are optimized one by one. Hence, (34) is decomposed into $K$ single-variable maximization sub-problems. In each inner iteration, the optimization problem can be casted as

where $r\in\left[{1,K}\right]$ and ${\bf{b}}_{r}^{\left({{{n}}+{{1}}}\right)}=(b_{1}^{({n+1})},\cdots,b_{r-1}^{({n+1})},{b_{r}},b_{r+1}^{(n)},\cdots,b_{K}^{(n)})$. However, it is still tremendously challenging to solve (34) that is an integer programming problem. Nevertheless, by relaxing the feasible region of $b_{r}$ to a real number, it is found that the outage probability $p_{out,k}$ is a convex function with respect to $b_{r}$ given the fixed values of $b_{1},\cdots,b_{r-1},b_{r+1},\cdots,b_{K}$, as proved in the following theorem.

With the finding in Theorem 3, it is obvious that the feasible region of the problem (34) is convex because of the convexity of ${{\tilde{p}_{out,K}}}({{\bf{b}}_{r}^{\left({{{n+1}}}\right)}})$ with respect to $b_{r}$. Moreover, since ${\tilde{T}}({{\bf{b}}_{{r}}^{({{{n}}+{{1}}})}})=\sum\nolimits_{k=1}^{K}{{N_{k}}{\tilde{p}_{out,k-1}}({{\bf{b}}_{{r}}^{({{{n}}+{{1}}})}})}$ is a summation of the convex functions, ${\tilde{T}}({{\bf{b}}_{{r}}^{({{{n}}+{{1}}})}})$ is also a convex function. However, $\tilde{\mathcal{R}}\left({{\bf{b}}_{{r}}^{\left({{{n}}+{{1}}}\right)}}\right)$ is not convex. To address this issue, successive convex approximation (SCA) can be used to relax (34) as a convex optimization problem. More specifically, by relying on the first-order Taylor approximation, $\tilde{\mathcal{R}}({{\bf{b}}_{r}^{\left({{{n}}+{{1}}}\right)}})$ is approximated as

where $\tilde{\bf{b}}_{r}^{\left({{{n}}+{{1}}}\right)}=(b_{1}^{({n+1})},\cdots,b_{r-1}^{({n+1})},{b_{r}^{({n})}},b_{r+1}^{(n)},\cdots,b_{K}^{(n)})$ and ${{p}_{out,k-1}^{\prime}}({{\bf{\tilde{b}}}_{r}^{(n+1)}})$ denotes the first-order partial derivative with respect to $b_{r}$. By substituting (III-B) into (34) leads to

Since the objective function of problem (36) is a concave function with respect to $b_{r}$ and the feasible region is also a convex set, the sub-problem (36) can be solved with the classical convex optimization tools. To sum up, the Dinkelbach’s algorithm is used to update $\alpha$ in the outer iterations until the difference between $\alpha^{(n+1)}$ and $\alpha^{(n)}$ is negligible or the maximum allowable number of iterations is reached. In the inner iterations, the SCA assisted alternating optimization algorithm is used to update $\bf b$.

For simplicity, we assume equal power allocation scheme, i.e., ${P_{1}}=\cdots={P_{K}}\triangleq P$. By using the definition of the EE in (37), the energy efficiency can be rewritten as

As proved in (15), the SE ${{{\cal T}_{K}}}$ is not greater than the EC $\bar{C}={\mathbb{E}}\{{{{\log}_{2}}({1+{{\left|h_{1}\right|}^{2}}P/\sigma^{2}})}\}$, that is, ${{\cal T}_{K}}\leq\bar{C}$. Accordingly, the EE is upper bounded as

where the equality holds if and only if the VL-XP-HARQ achieves the ergodic capacity. In addition, it is readily proved that the upper bound ${\mathbb{E}}\{{{{\log}_{2}}({1+{{\left|h_{1}\right|}^{2}}P/\sigma^{2}})}\}/P$ is a decreasing function of ${P}$. The maximum value of the upper bound is achieved at $P=0$, i.e., ${\mathbb{E}}\{{{{\log}_{2}}({1+{{\left|h_{1}\right|}^{2}}P/\sigma^{2}})}\}/P<{\mathbb{E}}\{{\left|h\right|^{2}}\}/(\sigma^{2}\ln 2)$. Hence, the maximum EE is bounded as

In this section, we focus on the maximization of the EE. More specifically, the transmission powers in each HARQ round are optimized to maximize the EE while maintaining an outage constraint (i.e., ${p_{out,K}}\leq\varepsilon$). The maximization of the EE is therefore formulated as

where ${\bf P}=(P_{1},\cdots,P_{K})$. Similarly, the asymptotic outage probability in Theorem 2 is used to replace the true outage probability in (41) facilitate the optimization. Based on the finding that (41) takes the similar form as the maximization of the SE in (18), (41) is thus rewritten by applying Dinkelbach’s transform as

where $\bar{\mathcal{R}}({\bf{P}})=\sum\nolimits_{k=1}^{K}{b_{k}}({\bar{p}_{out,k-1}}({\bf{P}})-{\bar{p}_{out,K}}({\bf{P}}))$, $\bar{\mathcal{E}}\left({\bf{P}}\right)=\sum\nolimits_{k=1}^{K}{{N_{k}}P_{k}{\bar{p}_{out,k-1}}\left({\bf{P}}\right)}$, ${\bar{p}_{out,k}}(\mathbf{P})$ refers to the asymptotic outage probability of VL-XP-HARQ with the transmission powers in different HARQ rounds set as $\bf P$ and ${\bar{p}_{out,k}}(\mathbf{b})$ can be evaluated through Theorem 2. Moreover, in the outer iterations, the Dinkelbach’s algorithm is used to update the auxiliary variable $\beta$ as

where $n$ the iteration index.

Similarly, the SCA aided alternating optimization algorithm is used to update $\bf P$ in the inner iterations. Thus, (41) can be divided into $K$ sub-problems and the first-order Taylor expansion can be leveraged to approximate $\bar{\mathcal{R}}\left({\bf{P}}\right)$. To proceed with the optimization, we define ${{\bf{P}}_{r}^{{{(n+1)}}}}=({P_{1}^{\left({n+1}\right)},\cdots,P_{r-1}^{\left({n+1}\right)},{P_{r}},R_{r+1}^{\left(n\right)},\cdots,P_{K}^{\left(n\right)}})$. Thereupon, each sub-problem reads as

where $\breve{\mathcal{R}}({{\bf{P}}_{r}^{(n+1)}})$ is given by (IV-B), as shown at the top of the next page, where $\tilde{\bf{P}}_{r}^{(n+1)}=(P_{1}^{\left({n+1}\right)},\cdots,P_{r-1}^{\left({n+1}\right)},P_{r}^{\left(n\right)},R_{r+1}^{\left(n\right)},\cdots,P_{K}^{\left(n\right)})$ and ${{\bar{p}}_{out,k-1}^{\prime}}({\tilde{\bf{P}}_{r}^{(n+1)}})$ denotes the first-order partial derivative with respect to $P_{r}$.