# Non-Orthogonal eMBB-URLLC Radio Access for Cloud Radio Access Networks with Analog Fronthauling
## 3 Analog Fronthaul Signal Processing
### Radio Resource Mapping over Fronthaul Channels  
To maximize the SNRs for all the signals forwarded over the fronthaul by symmetry, we need to ensure that: (i) all the received signals are replicated \(\eta\) times across the cable twisted-pairs, where we recall that \(\eta\) is the bandwidth amplification factor defined in (8); and (ii) cable cross-talk interference among different radio frequency bands is minimized. In fact, as the transmitted power at the cable input is limited by the constraints in (11), a simple and effective way to cope with the impairments of the analog fronthaul links using the only analog-processing capability is by introducing redundancy in the fronthaul transmission. To this end, without loss of generality, we assume the following mapping rule between the \(n_{F}\) radio signals at each EN and the \(l_{S}l_{F}\) cable resources. Let us consider the radio signal \(\mathbf{y}_{k}= \begin{bmatrix}Y_{k}^{1},&Y_{k}^{2},&...,&Y_{k}^{n_{F}}\end{bmatrix} ^ {T}\) received at the \(k\)-th EN. For a given normalized cable bandwidth \(\mu\), the \(n_{F}\) frequency channels of the radio signal \(\mathbf{y}_{k}\) can be split into \(1/\mu\) sub-vectors of size \(l_{F}=\mu n_{F}\) as  
\[\mathbf{y}_{k}= \begin{bmatrix}\mathbf{y}_{k}^{1}\\ \mathbf{y}_{k}^{2}\\ \vdots\\ \mathbf{y}_{k}^{\frac{1}{\mu}}\end{bmatrix} , \tag{15}\]  
where each vector \(\mathbf{y}_{k}^{j}\) contains a disjoint fraction of the radio signal bandwidth \(n_{F}\). Each vector \(\mathbf{y}_{k}^{j}\) can be transmitted over \(\eta\) twisted-pairs, where we recall that \(\eta\) is the fronthaul redundancy factor. To this end, each signal \(\mathbf{y}_{k}^{j}\) in (16) is transmitted over \(\eta\) consecutive cable twisted-pairs. To formalize the described mapping, the first step consists in reorganizing the signal \(\mathbf{y}_{k}\) into a \(l_{F}\times\frac{1}{\mu}\) matrix as  
\[\mathbf{Y}_{k}=\text{vec}_{\frac{1}{\mu}}^{-1}(\mathbf{y}_{k})= \begin{bmatrix} \mathbf{y}_{k}^{1}&\mathbf{y}_{k}^{2}&...&\mathbf{y}_{k}^{\frac{1}{\mu}}\end{bmatrix} , \tag{16}\]  
where the operator \(\text{vec}_{\frac{1}{\mu}}^{-1}(\cdot):\,\mathbb{C}^{n_{F}}\to\mathbb{C}^{n_{ F}\cdot\mu\times\frac{1}{\mu}}\) acts as the inverse of the vectorization operator \(\text{vec}(\cdot)\), with the subindex \(1/\mu\) denoting the number of columns of the resulting matrix. Then, the overall cable signal \(\mathbf{\hat{Y}}_{k}\) transmitted by EN \(k\)-th in (9) can be equivalently written as  
\[\mathbf{\hat{Y}}_{k}= \begin{bmatrix}\underbrace{\mathbf{y}_{k}^{1}&\mathbf{y }_{k}^{1}&\cdots&\mathbf{y}_{k}^{1}}_{\eta},&\underbrace{\mathbf{y}_{k}^{2}& \mathbf{y}_{k}^{2}&\cdots&\mathbf{y}_{k}^{2}}_{\eta}&...&\underbrace{\mathbf{y }_{k}^{\frac{1}{\mu}}&\mathbf{y}_{k}^{\frac{1}{\mu}}&\cdots&\mathbf{y}_{k}^{ \frac{1}{\mu}}\\ \end{bmatrix} , \tag{17}\]  
or, in compact form, as  
\[\mathbf{\hat{Y}}_{k}=\mathbf{Y}_{k}\otimes\mathbf{1}_{\eta}^{T}, \tag{18}\]  
where \(\otimes\) denotes the Kronecker product (for a review of Kronecker product properties in signal processing we refer the reader to [38]). Notice that in case of full normalized cable bandwidth, i.e., \(\mu=1\) (corresponding to \(\eta=l_{S}\)), the signal \(\mathbf{\hat{Y}}_{k}\) transmitted over the fronthaul cable simplifies to \(\mathbf{\hat{Y}}_{k}=\mathbf{y}_{k}\otimes\mathbf{1}_{l_{S}}^{T}\), which implies that the radio signal \(\mathbf{y}_{k}\) is replicated over all the \(l_{S}\) twisted-pairs. On the contrary, when the normalized cable bandwidth is minimal, i.e., \(\mu=1/l_{S}\) (corresponding to \(\eta=1\)), the signal \(\mathbf{\hat{Y}}_{k}\) does not contain any redundancy, and disjoints signals are transmitted over all pairs, so that the cable signal \(\mathbf{\hat{Y}}_{k}\) equals the matrix radio signal in (16) as \(\mathbf{\hat{Y}}_{k}=\mathbf{Y}_{k}\). **Remark--Practical Implementation Issues.