# Protograph LDPC Code Design for Asynchronous Random Access +
## 3 Code Design for the Asynchronous Random Access Channel
### Simplified Channel Models for Code Design
#### 3.3.2 Single Interferer Model  
The Gaussian interference model is a reasonable approximation already in the presence of a few interferers under the assumption of time, frequency, and phase offset. When a single interferer is present over the codeword, the Gaussian model may become imprecise, and the interference shall be better characterized, taking into account the chosen modulation. Thus, in this Section, we consider a codeword affected by interference generated by a single QPSK-modulated interferer, received with the same power. After ideal detection and ideal channel estimation, the interference occurring on a codeword symbol \(i\) presents a relative phase shift \(\Delta\varphi^{(1)}\) and a relative epoch \(0<\Delta\varepsilon^{(1)}<t_{s}\). Hence, the interference on codeword symbol \(i\), i.e., \(z_{i}\), can be expressed as:  
\[z_{i}=\hat{x}^{(1)}(kt_{s}-\Delta\varepsilon^{(1)}-\Delta t_{0}^{(1)})e^{j \Delta\varphi^{(1)}}.\]  
Note that \(\gamma_{i}=\mathsf{P}/\mathsf{Z}_{i}=1\). For the ease of notation, we drop the superscript \({}^{(1)}\), so that \(\Delta\varphi^{(1)}\triangleq\Delta\varphi\) and \(\Delta\varepsilon^{(1)}\triangleq\Delta\varepsilon\). Figure 3 shows the considered scenario. The \(i^{\text{th}}\) codeword symbol is interfered by a portion \(\Delta\varepsilon\) of symbol duration with the \((k-1)^{\text{th}}\) symbol of the interferer and by \(t_{s}-\Delta\varepsilon\) with the \(k^{\text{th}}\) symbol of the interferer. We define \(\mathcal{S}=\{S_{1},S_{2},S_{3},S_{4}\}\) as the set of QPSK constellation points with respective labels \(\{00,01,11,10\}\). For a given relative epoch \(\Delta\varepsilon\) and a relative phase offset \(\Delta\varphi\), known at the receiver, the channel LLR of Bit Level 1 of a Gray-labelled QPSK signal is given by:  
\[L_{1}(y,\Delta\varphi,\Delta\varepsilon^{\prime})=\log\left[\frac{\sum_{\{S_{k -1},S_{k}\}\in\mathcal{S}^{2}}\left(e^{-\frac{\left|y-S_{1}-e^{j\Delta\varphi }(\Delta\varepsilon^{\prime}S_{k-1}+(1-\Delta\varepsilon^{\prime})S_{k}) \right|^{2}}{2\pi_{n}^{2}}}+e^{-\frac{\left|y-S_{2}-e^{j\Delta\varphi}(\Delta \varepsilon^{\prime}S_{k-1}+(1-\Delta\varepsilon^{\prime})S_{k})\right|^{2}} {2\pi_{n}^{2}}}\right)}{\sum_{\{S_{k-1},S_{k}\}\in\mathcal{S}^{2}}\left(e^{- \frac{\left|y-S_{3}-e^{j\Delta\varphi}(\Delta\varepsilon^{\prime}S_{k-1}+(1- \Delta\varepsilon^{\prime})S_{k})\right|^{2}}{2\pi_{n}^{2}}}+e^{-\frac{ \left|y-S_{4}-e^{j\Delta\varphi}(\Delta\varepsilon^{\prime}S_{k-1}+(1-\Delta \varepsilon^{\prime})S_{k})\right|^{2}}{2\pi_{n}^{2}}}\right)}\right] \tag{4}\]where we defined \(\Delta e^{\prime}\triangleq\Delta e/t_{s}\). For the sake of simplification, we assume symbol-synchronous interference, i.e., \(\Delta e=\Delta e^{\prime}=0\), and unknown, uniformly-distributed relative phase offset. We can write the the channel LLR of Bit Level 1 as:  
\[L_{1}(y,\Delta e=0)=\log\left[\frac{\sum_{S_{1}\in\mathcal{S}}\int_{-\pi}^{\pi} \left(e^{-\frac{\left|y-S_{1}-S_{2}e^{i\Delta\varphi}\right|^{2}}{2\pi_{1}^{2} }}+e^{-\frac{\left|y-S_{2}-S_{2}e^{i\Delta\varphi}\right|^{2}}{2\pi_{1}^{2}}} \right)d\theta}{\sum_{S_{1}\in\mathcal{S}}\int_{-\pi}^{\pi}\left(e^{-\frac{ \left|y-S_{3}-S_{4}e^{i\Delta\varphi}\right|^{2}}{2\pi_{1}^{2}}}+e^{-\frac{ \left|y-S_{4}-S_{4}e^{i\Delta\varphi}\right|^{2}}{2\pi_{1}^{2}}}\right)d\theta }\right]. \tag{5}\]  
Figure 4 shows the possible received symbol \(i\) after phase compensation with no noise when symbol \(S_{1}\) (or \(S_{2},S_{3},S_{4}\)) is transmitted. The received symbol can lay anywhere on the red circle if no assumption on interference relative phase shift is done; see Figure 4a. If a relative shift of \(\Delta\varphi=10^{\circ}\) is present, then any of the four red points can be received depending on the transmitted interference symbol \(k\); see Figure 4b. Figure 4: QPSK-modulated constellation after phase compensation at the receiver side and single symbol-synchronous interferer with the same power, relative phase shift, and no noise.