# Protograph LDPC Code Design for Asynchronous Random Access +
## 4 Numerical Results
### Abstracted Physical Layer
#### 4.1.1 Decoding Region for Random Code Ensembles  
We assumed that replicas are received with interference whose power may vary over the replica symbols. We resorted to a _block interference channel_[41], where a replica experiences various blocks of constant interference. Recall that the interference was assumed to be drawn from a complex white Gaussian process \(\mathcal{CN}(0,2\sigma_{1}^{2})\), with \(2\sigma_{1}^{2}\) being the interference power (see also Section 3.3.1).  
Let us call \(m^{\prime}\) the maximum number of interferers affecting a replica. Let us denote by \(m\), \(1\leq m\leq m^{\prime}+1\), the number of _different_ interference plus noise levels that are present over the considered replica. Define:  
\[\boldsymbol{\sigma}^{(2)}=\Big{[}\underbrace{\sigma_{n}^{2}+\frac{1}{2}}_{ \sigma_{1}^{2}},\ldots,\underbrace{\sigma_{n}^{2}+\frac{m}{2}}_{\sigma_{n}^{2 }}\Big{]} \tag{9}\]  
as the ordered interference plus noise vector, which contains the \(m\) different noise plus interference levels in the replica. Let \(\boldsymbol{\alpha}=[\alpha_{1},\ldots,\alpha_{m}]\), where \(\alpha_{j},j\in\{1,\ldots,m\}\), is the fraction of a replica subject to the interference plus noise of power \(\left(\sigma_{n}^{2}+\frac{j}{2}\right)\) per dimension. Clearly, \(\sum_{j=1}^{m}\alpha_{j}\leq 1\) and \(\alpha_{j}\in(0,1]\).  
Recalling that \(C\left(\sigma^{2}\right)\) is the QPSK constraint AWGN channel capacity for a given \(\sigma^{2}\), the outage capacity \(C_{o}\) under the Gaussian interference assumption is:  
\[C_{o}(\mathbf{\alpha},\sigma_{n}^{2},\sigma^{(2)})=\left(1-\sum_{j=1}^{m}\alpha_{j} \right)\,C\left(\frac{1}{2\sigma_{n}^{2}}\right)+\alpha_{1}\,C\left(\frac{1}{2 \sigma_{n}^{2}}\right)+\ldots+\alpha_{m}\,C\left(\frac{1}{2\sigma_{m}^{2}} \right).\]  
For a fixed transmission rate \(R\), we define the decoding region \(\mathcal{D}\)[33; 44] as:  
\[\mathcal{D}=\left\{\mathbf{\sigma}^{(2)}\in\mathbb{R}_{+}^{m},\mathbf{\alpha}\in(0,1]^ {m}\ |\ \sum_{j=1}^{m}\alpha_{j}\leq 1\wedge R<C_{o}(\mathbf{\sigma}^{(2)},\sigma_{n}^{2}, \mathbf{\alpha})\right\}, \tag{10}\]  
Every \(m^{\prime}\)-user collision resulting in \((\mathbf{\alpha},\sigma_{n}^{2},\mathbf{\sigma}^{(2)})\in\mathcal{D}\) can be resolved.  
A possibility to evaluate numerically the performance of a random code ensemble that achieves a fraction \(\beta\) of the outage capacity with \(\beta<1\), is to compute the decoding region \(\mathcal{D}^{\prime}\subset\mathcal{D}\) as:  
\[\mathcal{D}^{\prime}=\left\{\mathbf{\sigma}^{(2)}\in\mathbb{R}_{+}^{m},\mathbf{\alpha }\in(0,1]^{m}\ |\ \sum_{j=1}^{m}\alpha_{j}\leq 1\wedge R<\beta\,C_{o}(\mathbf{ \sigma}^{(2)},\sigma_{n}^{2},\mathbf{\alpha})\right\}.\]  
For the abstracted physical layer simulations, we followed the steps from the literature [44]. We generated realizations of packet collisions for a certain channel load \(G\). For each of the \(n_{s}\) codeword symbols of a replica, we computed an instantaneous noise plus interference power. We assumed that for each codeword symbol, we had knowledge of the number of interferers due to ideal detection. Thus, we grouped the \(n_{s}\) instantaneous noise plus interference power values into \(m\) blocks, sorted them according to the number of interferers, and averaged over the noise plus interference power in every block to finally obtain the ordered interference plus noise vector \(\mathbf{\sigma}^{(2)}\) in (9). Successful decoding was declared if \((\mathbf{\alpha},\sigma_{n}^{2},\mathbf{\sigma}^{(2)})\in\mathcal{D}\) (or \(\in\mathcal{D}^{\prime}\)).