# Protograph LDPC Code Design for Asynchronous Random Access +
## 3 Code Design for the Asynchronous Random Access Channel
### Asymptotic Results for a Single Non-Gaussian Interferer  
Although our ad-hoc protograph ensemble was designed for the Gaussian interference model, we show here that its asymptotic performance is also good in this case of non-Gaussian interferers. We consider the iterative decoding threshold of our ad-hoc protograph ensemble for two cases:  
Figure 6: Maximum interference power versus \(\alpha\) for different protographs and outage capacity on a QPSK channel with fixed AWGN at \(E_{s}/N_{0}=6\) dB. The 5G corresponds to the ensemble with base matrix \(\mathbf{B}_{\text{5G}}^{\pi}\). The 5G perm.corresponds to the ensemble with base matrix \(\mathbf{B}_{\text{5G}}^{\pi}\). Ad-hoc corresponds to the ensemble with base matrix \(\mathbf{B}_{\text{A}}\). Begin and end represent the iterative decoding threshold with the interference hitting the codeword from the left or the right, respectively.  
* Symbol-synchronous, phase-aligned, equal-power QPSK interferer, i.e., \(\Delta e=0\), \(\Delta\varphi=0\), \(\mathrm{P}=\mathrm{Z}=1\).
* Symbol-synchronous, equal-power QPSK interferer uniform at random phase, i.e., \(\Delta e=0\), \(\Delta\varphi\sim\mathcal{U}\left[0;2\pi\right)\), \(\mathrm{P}=\mathrm{Z}=1\).  
For reference, we also show the results for a single equal-power interferer if that interferer is assumed to be Gaussian. We determine the iterative decoding threshold of our ad-hoc protograph ensemble for these interference models by applying (quantized) density evolution, making use of Equations (4) and (5). Since the interferer power is fixed, we define the iterative decoding threshold as the largest noise power per dimension \(\sigma_{n}^{2}\) such that error-free transmission is possible. Figure 7 shows the density evolution threshold of the ad-hoc protograph ensemble compared to the channel capacity in each of the three interferer models.  
As shown in Figure 7, the QPSK interferer with random phase gives the best threshold, whereas the Gaussian interferer gives the worst threshold. We can explain this by using Figure 4. In the absence of noise, the received signal \(y(t)\) is on a circle around the transmitted signal, \(x(t)\). To depict \(y(t)\), we can simply draw four circles around the four QPSK constellation points. It can be observed that these circles intersect. However, the probability to pick exactly the intersection point goes to zero. Thus, we can reach a mutual information of two for the QPSK interferer with random phase. For the phase-aligned QPSK interferer, the mutual information between \(y(t)\) and \(x(t)\) reaches one in the absence of noise. Thus, for the interfered part, we can get a rate of at most one. Hence, the phase-aligned QPSK interferer gives a lower threshold than the QPSK interferer with random phase. For the Gaussian interferer case, the mutual information is limited by the QPSK capacity at \(E_{s}/N_{0}=0\)  
Figure 7: The noise threshold for the code ensemble described by the base matrix \(\mathbf{B_{A}}\), calculated using quantized density evolution, compared to capacity, on a QPSK channel with a single equal power interferer as the interferer overlap, \(\alpha\), is varied. Three different models for the interferer are considered, a time- and phase-aligned QPSK signal, a time-aligned and random phase QPSK signal and a Gaussian interferer.  
dB, where the mutual information at 0 dB is 0.96 for the interfered part. Thus, the Gaussian interference model shows the worst threshold compared to the phase-aligned QPSK interferer and QPSK interferer with random phase.