# Low-Latency Short-Packet Transmission over a Large Spatial Scale
## 2 System Model  
We focus on an orbital wireless system with the requirement of extremely low latency, which is demonstrated in Figure 1. The satellites communicate with another planet, e.g., Mars, via the space station. Each of the satellites is considered as a DF relay. For simplicity, we only consider the transmission of one packet from the source to destination, as shown in Figure 2, where a source node \(R_{0}\) transmits a \(k\)-bit packet in each transmission to its destination node \(R_{L}\) with the help of \(L-1\) decode-and-forward (DF) relays, called \(R_{1}\), \(R_{2},\ldots,R_{L-1}\). The distance between node \(R_{t-1}\) and \(R_{t}\) is \(d_{t}\). The \(k\)-bit packet is encoded into symbols to be transmitted from \(R_{0}\) to \(R_{L}\). The modulated waveforms of those symbols are reliably sent by node \(R_{t-1}\). Node \(R_{t}\) receives and detects it without distortion. We denote by \(\tau\) the average time consumed in sending one symbol. As a result, \(\frac{1}{\tau}\) is the symbol rate. Let \(n_{t}\) be the number of symbols transferred between node \(R_{t-1}\) and \(R_{t}\), \(n_{t}\in\mathbb{N}_{+}\). We assume node \(R_{t-1}\) encodes the \(k\)-bit packet into \(n_{t}\) symbols. \(R_{t-1}\) sends a symbol vector \(\mathbf{x}_{t}\) to \(R_{t}\). In particular, \(\mathbf{x}_{t}=(x_{t1},x_{t2},\cdots,x_{m_{t}})\) and \(|x_{t}|^{2}_{2}=1\). \(x_{ij}\) is the \(j\)th symbol. Let \(h_{t}\) denote the channel coefficient between node \(R_{t-1}\) and \(R_{t}\). Let \(g_{t}\) denote the channel gain, namely \(g_{t}=|h_{t}|^{2}\). We denote by \(\mathbf{y}_{t}\) the received symbol vector at node \(R_{t}\). Specifically, \(\mathbf{y}_{t}=(y_{t1},y_{2},\cdots,y_{m_{t}})\). For \(1\leq j\leq n_{t}\), \(y_{ij}\) can be presented by:  
\[y_{ij}=h_{t}x_{ij}+z_{t}, \tag{1}\]  
where \(z_{t}\) is the additive Gaussian white noise. Let \(d_{t}\) denote the distance between node \(R_{t-1}\) and \(R_{t}\). The path loss of the transmission from \(R_{t-1}\) to \(R_{t}\) is normalized as \(d_{t}^{-\alpha}\), where \(\alpha\) is the loss exponent. Due to the large scale of our model, the shadow-fading component can be ignored, and the channel gain is presented as \(g_{t}=d_{t}^{-\alpha}\)[24]. We assume the signal-to-noise ratio (SNR) of the signal emitted by \(R_{t-1}\) is \(\gamma_{t}\). Thus, node \(R_{t}\) receives the signal with the SNR equal to \(\gamma_{t}d^{-\alpha}\) and sends it to the next node \(R_{t+1}\) with the SNR equal to \(\gamma_{t+1}\) if \(R_{t}\) detects no error. We then present the overall reliability from the source to the destination. To achieve extremely low latency and ultrareliability, a finite-blocklength coding scheme is adopted. For the real AWGN channel, a tight bound on the coding rate was given by [6]. Let \(\epsilon_{t}\) denote the error probability from node \(R_{t-1}\) to \(R_{t}\). Since \(R_{t-1}\) transmits \(k\) bits on \(n_{t}\) symbols, the coding rate is \(\frac{k}{n_{t}}\). Using the normal approximation in this paper, we have the formula for the channel coding rate in the transmission from \(R_{t-1}\) and \(R_{t}\), i.e.,  
\[\frac{k}{n_{t}}=C_{t}-\sqrt{\frac{V_{t}}{n_{t}}}Q^{-1}(\epsilon_{t}), \tag{2}\]  
in which \(Q(x)=\int_{x}^{\infty}\frac{1}{\sqrt{2\pi}}\exp(-\frac{t^{2}}{2})dt\) is the Gaussian Q-function. \(Q^{-1}\) is the inverse function of \(Q(x)\). We denote by \(C_{t}\) and \(V_{t}\) the channel capacity and the channel diversity, respectively. Under our configuration, the channel capacity is represented by:  
\[C_{t}=\log(Substituting Equation (3) into Equation (2), we present the error probability \(\epsilon_{i}\):  
\[\epsilon_{i}=Q\Bigg{(}\left(\ln(1+\gamma_{i}d_{i}^{-\alpha})-\frac{k\ln 2}{n_{i}} \right)\frac{n_{i}}{V_{i}}\Bigg{)}. \tag{5}\]  
Figure 1: An orbital relaying system. Figure 2: System model: a communication system with \(L-1\) relays. We assume all relay nodes and the destination node can decode the data packets and detect the error reliably. The relay nodes will forward the messages if and only if no error occurs.