# Intelligent Radio Frequency Identification for URLLC in Industrial IoT Networks
## 4 Transmission Performance Analysis of RFI-Aided URLLCs with Massive MIMO  
### Error Probability Bound for Random Channels  
In this section, we consider a simple case where the channel is assumed to have Gaussian distribution. Then, we establish an evaluation framework of URLLCs error probability with finite block-length uplink transmission. Due to the symmetry of uplink and downlink transmissions, our paper discusses uplink transmission performance. For simplicity but without loss of generality, the performance downlink is similar to uplink transmission. Furthermore, we involve massive MIMO into the system to assess the suitability for short-packet URLLCs. A typical channel model can be expressed as  
\[y(m)=\mathbf{h}Gs(m)+z(m)=\mathbf{h}x(m)+z(m),\quad m=1,\ldots,n \tag{21}\]  
where \(x(m)\in\mathbb{C}\) and \(y(m)\in\mathbb{C}\) represent the input and output of channel, respectively. \(\mathbf{h}\) denotes the channel gain, which is assumed to be a quasistatic variable during packet transmission. \(z(m)\) is a sample of additive Gaussian noise \(\mathcal{CN}\big{(}0,\sigma^{2}\big{)}\) that satisfies the  
Figure 6: Illustration of one-dimensional RFIResNet architecture used in this paper. independent and identically distributed (i.i.d) property. Then, we can obtain the estimated \(\widehat{x}\) by  
\[\widehat{x}=\operatorname*{arg\,min}_{\widehat{x}\in\mathcal{C}}\lVert y-\widehat {h}\widetilde{x}\rVert^{2} \tag{22}\]  
where \(\widetilde{x}\) denotes the predefined codeword, and \(\widehat{h}\) is the estimation of \(h\) at the receiver. Equation (22) shows that it is a classical decoder to estimate an original signal by minimizing Euclidean distance [38]. Then, we can obtain the error probability bound by  
\[ \begin{split} P_{Error}&=\mathbb{P}[\widehat{x} \
eq x]\\ &\leq\mathbb{P}\Bigg{[}\sum_{m=1}^{n}\gamma_{s}(x(m),y(m))+\log(u) \leq\log(k-1)\Bigg{]}\end{split}  \tag{23}\]  
where \(u\) is a uniformly distributed random variable, and \(\gamma_{s}\) represents information density, which is defined as follows [16]:  
\[\gamma_{s}(x(m),y(m))=-s\mid y(m)-\widehat{h}x(m)\Big{|}^{2}+\frac{s|y(m)|^{2}} {1+s\rho|\widehat{h}|^{2}}+\log\Bigl{(}1+s\rho|\widehat{h}|^{2}\Bigr{)} \tag{24}\]  
where \(\rho\) is considered to be the transmission power of an average packet. Due to random channel gain \(h\in\mathbb{C}\) satisfying an arbitrary distribution, we can rewrite (23) as  
\[ \begin{split} P_{Error}&=\mathbb{P}[\widehat{x} \
eq x]\\ &\leq\mathbb{E}_{h,\widehat{h}}\Bigg{[}\mathbb{P}\left[\sum_{m=1} ^{n}\gamma_{s}(x(m),y(m))\leq\log\frac{k-1}{u}\mid h,\widehat{h}\right]\Bigg{]} \end{split}  \tag{25}\]  
where the average is taken over the random distribution of \(h\). For general applications of infinite block-length transmission, Shannon capacity is the upper bound of the whole total system [13]. Furthermore, the error probability of URLICs can be defined as the probability that the empirical average of generalized information density \(\gamma_{s}\) described in (23) is smaller than chosen rate \(R\). As introduced in (25), an accurate closed-form mathematical solution can only be calculated by numerical simulations. Normal approximation (NA) is an efficient method to solve the huge calculation-consuming problem by utilizing the Berry-Esseen central limit theorem. On this basis, we can rewrite (25) as  
\[\mathbb{P}\left[\sum_{m=1}^{n}\gamma_{s}(x(m),y(m))\leq\log\frac{k-1}{u}\right] = Q\left(\frac{nI_{s}-\log(k-1)}{\sqrt{nI_{s}}}\right)+o\left(\frac{1}{\sqrt{ n}}\right) \tag{26}\]  
where \(I_{s}\) denotes mutual information, and \(Q\) represents Gaussian \(Q\) function. \(J_{s}\) defines the variance in information density. However, the approximation accuracy of NA is not robust for URLLC applications. Saddle-point approximation is a novel method to solve the huge calculation-consuming problem [39]. By using saddle-point approximation, we can obtain new representation of error probability. \[