# Protograph LDPC Code Design for Asynchronous Random Access +
## 2 System Model
### Asynchronous Random Access Protocol  
We consider an uncoordinated asynchronous random access protocol based on [25; 26]. With respect to [25], no concept of the MAC frame is present, so that the terminals operate in a full asynchronous scenario. Compared to [26], local time slots constraining the transmission delay between replicas of the same user are eliminated. An infinite user population generates data traffic modelled as a Poisson process with intensity \(\mathsf{G}\), called the _channel load_. It is measured in packet arrivals per packet duration \(t_{p}\). Specifically, the probability distribution that \(u\) users initiate a transmission within a packet duration is \(P(u)=\frac{\mathsf{G}^{\prime}e^{-\mathsf{G}}}{ut!}\). Note that throughout the paper, we use the convention of writing the exponential \(\exp\) as \(e\). Users are allowed to make a single attempt to transmit their data, i.e., no re-transmissions are considered. Prior to transmission, the data packet is replicated \(d=2\) times. We refer to each copy as a _replica_. The replicas are transmitted adhering to the following rules:  
1. self-interference must be avoided, i.e., no portion of the two replicas shall overlap;
2. the maximum delay, called virtual frame (VF), between the start of the first replica and the end of the second one shall not exceed \(t_{f}\) seconds;
3. the delay between the start of the two replicas is drawn uniformly at random within the interval \(\left(t_{0}^{(u)}+t_{p}\), \(t_{0}^{(u)}+t_{f}-t_{p}\right]\), with \(t_{0}^{(u)}\) the activation time of User \(u\). Once the transmission times for the replicas are chosen, this information is stored in the header in order to enable SIC at the receiver. Then, the replicas are encoded so as to be protected against noise and multiple access interference. At the receiver side, the incoming signal is sampled, stored and subsequently processed. The receiver makes use of a decoding window of size \(W\) samples. At first, replicas are detected, e.g., with correlation-based rules [35; 36]. In this work, ideal detection is considered, i.e., all transmitted replicas can be detected by the receiver, and no false alarms are present. Channel estimation is performed (for details, see Section 2.2). The soft-demodulator provides bit-wise LLRs as input to the channel decoder. When the cyclic redundancy check (CRC) matches, the replica is declared as correctly decoded. When this is the case, readily, two operations follow:  
1. the replica waveform is reconstructed on a sample level and subtracted from the incoming signal. We will consider ideal interference cancellation, i.e., after cancellation, no residual power is left by the replica;
2. the information about the twin location (the time position of the other replica of the same user) is extracted from the header. The second operation is followed by data-aided channel estimation on the packet copy. In fact, all the data carried by the packet are now known and can be used as pilots for refining channel estimation. The received waveform of the twin can be reconstructed and removed from the received signal. We shall note that if a replica can be decoded, the interference reduction triggered by SIC may impact underlying collided packets and can lead to further packet recovery. The receiver proceeds with extracting the second candidate replica and repeats the aforementioned operations. SIC is iterated until no more packets are present in the decoding window or when a predefined maximum number of iterations is reached. The second option is normally adopted when the receiver has tighter complexity or delay constraints. Once the operations on the current receiver decoding window are terminated, the receiver window is shifted forward by \(\Delta W\ll W\) samples, and the detection of the replicas is initiated again. **Example 1**. : _Consider an instance of SIC as depicted in Figure 1. A SIC step consists of successfully decoding a replica and removing it, as well as its twin from the received signal.