# Intelligent Radio Frequency Identification for URLLC in Industrial IoT Networks
## 4 Transmission Performance Analysis of RFI-Aided URLLCs with Massive MIMO
### Reliability Analysis for RFI-Aided URLLCs  
In this section, a single cell where the BS is equipped with \(M\) antennas serving two single-antenna smart devices is considered. A Rayleigh fading channel is involved to characterize the actual channel. We used \(\mathbf{h}\) to denote the channel vector between BS and smart device. Wireless transmission relied on pilot \(\ell_{pilot}\in\mathbb{C}\), and channel correlation directly affects BS performance. We used orthogonal pilot sequences to reduce mutual interference between different devices as much as possible. That means that \(\ell_{pilot}\) satisfies \(\ell_{pilot}^{i}\times\ell_{pilot}^{j}=0\), and the element pilot sequence is scaled by transmission power \(\sqrt{\rho_{ul}}\). At the receiver side, received pilot signal \(y_{pilot}\in\mathbb{C}\) can be expressed as  
\[y_{pilot}=\sqrt{\rho_{ul}}h_{i}\ell_{i}^{H}+\sqrt{\rho_{ul}}h_{j}\ell_{j}^{H} +z_{pilot} \tag{31}\]  
where \(z_{pilot}\in\mathbb{C}\) denotes the additive Gaussian noise with distribution \(\mathcal{CN}\left(0,\sigma^{2}\right)\). \(h_{i}\) is the fading channel vector. Furthermore, for accurately recovering the original signal on the receiving side, it is necessary to accurately estimate channel vector \(h\). Minimal mean squared error (MMSE) is a remarkable method to estimate \(h\) by  
\[\widehat{h}_{i}=\sqrt{\rho_{ul}}n_{p}\mathbf{R}_{i}\mathbf{W}_{i}^{-1}\Big{(} y_{pilot}\ell_{i}\Big{)} \tag{32}\]  
where  
\[\mathbf{W}_{i}=\sum_{l=1}^{K}\rho_{ul}\mathbf{R}_{i}\ell_{l}^{H}\ell_{i}+ \sigma^{2}\mathbf{I}_{M} \tag{33}\]  
and \(K=2\) denotes two different smart devices. \(\mathbf{R}\) is the correlation matrix of spatial channel, which indicates the quality of different channels. It is clear from (33) that, if pilot sequences are orthogonal, i.e., \(\ell_{j}^{H}\ell_{i}=0\), there is no interference in channel estimation. Furthermore, if underlying channels \(h_{1}\) and \(h_{2}\) are independent from each other, this means that \(h_{1}h_{2}^{H}=0\). Then, channel gain is independent scaling factor \(r_{i}\), which is the diagonal element of \(\mathbf{R}=\beta_{i}\mathbf{I}_{M}\). Uplink transmission can be described as  
\[y_{ul}(m)=\sum_{i=1}^{K}\mathbf{h}_{i}x_{ul,i}(m)+z_{ul} \tag{34}\]  
where \(x(m)\) denotes the original signal that is transmitted by the \(i\)-th smart device with average power \(\rho_{ul}\). \(z_{ul}\) is the Gaussian additive noise satisfying i.i.d properties. We can estimate \(\widehat{x}_{ul}\) through estimation-combined channel vector channel \(V_{i}\), which can be obtained by computational algorithm MMSE in (35). \[\mathbf{V}_{i}=\left(\sum_{i=1}^{k}\widehat{h}_{i}\widehat{h}_{i}^{H}+Z\right)^ {-1}\widehat{h}_{i} \tag{35}\]  
where \(Z=\sum_{i=1}^{k}\Phi_{i}+\frac{\sigma_{ul}^{2}}{\rho_{ul}}\mathbf{I}_{\text{M}}\) denote mutual interference between different devices. Furthermore, we can obtain signal \(x_{ul,i}(m)\) by utilizing combined vector \(V_{i}\) as follows:  
\[r_{ul,i}^{\text{ul}}[k]=V_{i}^{H}\sum_{i=1}^{K}\mathbf{h}_{i}x_{ul,i}(m)+V_{i} ^{H}z_{ul} \tag{36}\]  
Similar to (21), (36) has the same form. Assuming that BSs achieve a perfect estimation of channel can obtain the final codeword by  
\[\widehat{x}_{ul,i}=\underset{\widetilde{x}_{ul,i}\in C_{ul}}{\arg\min}\Big{\|} r_{ul,i}-\left(V_{i}^{H}\widehat{h}_{i}\right)\widetilde{x}_{ul,i}\Big{\|}^{2} \tag{37}\]  
To obtain the average error probability of uplink transmission, we refer to the calculation process of (23). A simplified expression can be written as  
\[P_{ul,i}\leq\mathbb{E}\Bigg{[}\mathbb{P}\Bigg{[}\sum_{i=1}^{n_{ul}}\gamma_{s }(r_{ul,i}(m),x_{ul,i}(m))\leq\log\frac{k-1}{u}\mid V_{i}^{H}h_{i},V_{i}^{H} \widehat{h}_{i},\sigma^{2}\Bigg{]}\Bigg{]} \tag{38}\]  
where \(n_{ul}l\) represents the number of uplink transmissions. The saddle-point approximation described in Section 4.1 can be used to efficiently and accurately calculate probability in (38).