# Protograph LDPC Code Design for Asynchronous Random Access +
## 2 System Model
### Asynchronous Random Access Channel Model  
In the following, we define the general channel model for the considered random access scheme. Assume that all replicas of all users form the set \(\mathcal{R}\). Consider the \(r^{\text{th}}\) replica in \(\mathcal{R}\). The transmitted signal \(x^{(r)}\) corresponding to the modulated codeword \(\mathbf{a}^{(r)}=\left(a_{0}^{(r)},a_{1}^{(r)},\ldots,a_{n_{s}-1}^{(r)}\right)\) is:  
\[x^{(r)}(t)=\sum_{i=0}^{n_{s}-1}a_{i}^{(r)}g(t-it_{s}),\]  
where \(n_{s}\) is the number of modulation symbols in a packet, \(t_{s}\) is the symbol duration, and \(g(t)\) is the pulse shape. The received signal is in general affected by a frequency offset, modelled as a uniformly-distributed random variable \(f^{(r)}\sim\mathcal{U}\left[-f_{\text{m}};f_{\text{m}}\right]\), with \(f_{\text{m}}\) the maximum frequency offset. A sampling epoch is also modelled as a uniformly distributed random variable \(\epsilon^{(r)}\sim\mathcal{U}\left[0;t_{s}\right)\) ([37], Chapter 2). Both frequency offset and sampling epoch are common to each replica of the same user, but independently user-by-user. Furthermore, the signal is also affected by phase offset \(\varphi^{(r)}\), modelled as a uniformly-distributed random variable between zero and \(2\pi\), i.e., \(\varphi^{(r)}\sim\mathcal{U}\left[0;2\pi\right)\). Differently from the frequency offset and epoch, the phase offset is assumed to be independent replica-by-replica, i.e., \(\varphi\) may vary even among replicas of the same user. Further, we assume AWGN. No fading is  
Figure 1: Example of the collision pattern at the receiver and of the SIC procedure. Upon correct decoding, SIC removes the packet and its twin. considered, which is typical, e.g., for fixed-terminal geostationary orbit satellite scenarios. For \(f_{\text{m}}t_{s}\ll 1\), the received signal \(y(t)\) after matched filtering, which is a superposition of all replicas \(r\in\mathcal{R}\), is:  
\[y(t)=\sum_{r\in\mathcal{R}}\tilde{x}^{(r)}(t-\epsilon^{(r)}-t_{0}^{(r)})e^{i \left(2\pi f^{(r)}+\varphi^{(r)}\right)}+n(t), \tag{1}\]  
where \(\tilde{x}^{(r)}(t)\) is the matched filtered signal \(x^{(r)}(t)\). In Equation (1), \(t_{0}^{(r)}\) is the \(r\)th replica delay w.r.t. the common reference time. The noise term \(n(t)\) is given by \(n(t)\triangleq\
u(t)*g(t)\), where \(\
u(t)\) is a white Gaussian process with single-sided power spectral density \(N_{0}\). For the sake of simplicity, throughout the paper, we make the assumptions that \(f^{(r)}=0\) and \(\epsilon^{(r)}=0\), \(\forall r\in\mathcal{R}\). Hence, Equation (1) becomes:  
\[y(t)=\sum_{r\in\mathcal{R}}\tilde{x}^{(r)}(t-t_{0}^{(r)})e^{i\varphi^{(r)}}+n (t).\]  
We focus again on replica \(r\). After ideal detection, the received signal corresponding to \(r\) can be identified and isolated. Ideal channel estimation recovers perfectly the phase offset \(\varphi^{(r)}\). After compensation, the discrete-time version of the received signal \(\mathbf{y}^{(r)}=(y_{0}^{(r)},\ldots,y_{n_{s}-1}^{(r)})\) corresponding to the replica \(r\) is given by:  
\[\mathbf{y}^{(r)}=\mathbf{a}^{(r)}+\mathbf{z}^{(r)}+\mathbf{n}^{(r)}, \tag{2}\]  
here, \(\mathbf{n}^{(r)}=(n_{0}^{(r)},\ldots,n_{n_{s}-1}^{(r)})\) are the samples of a complex discrete white Gaussian process with \(n_{i}^{(r)}\sim\mathcal{CN}(0,2\sigma_{n}^{2})\), \(\forall i\in\{0,\ldots,n_{s}-1\}\), \(\forall r\in\mathcal{R}\). The aggregate interference contribution over the replica-\(r\) signal is \(\mathbf{z}^{(r)}=\left(z_{0}^{(r)},\ldots,z_{n_{s}-1}^{(r)}\right)\), with:  
\[z_{i}^{(r)}=\sum_{r\in\mathcal{R}r}\tilde{x}^{(r)}(tk_{s}-\Delta t_{0}^{(r)}) e^{i\Delta\varphi^{(r)}}\qquad\text{for all $i\in\{0,\ldots,n_{s}-1\}$},\]  
here, \(\Delta t_{0}^{(r)}=t_{0}^{(r)}-t_{0}^{(r)}\) and \(\Delta\varphi^{(r)}=\varphi^{(r)}-\varphi^{(r)}\). The instantaneous received power for symbol \(i\) is \(\mathsf{P}_{i}^{(r)}\triangleq\mathbb{E}\left[|a_{i}^{(r)}|^{2}\right]\).