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Protograph LDPC Code Design for Asynchronous Random Access^{ †}

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## Abstract

**:**

## 1. Introduction

- In Section 3, we present a surrogate channel model, exploited in the code design phase, which assumes constant interference power over a fraction $\alpha $ of the codeword. To facilitate code design, we further approximate the aggregate interference contribution, possibly generated by a multitude of terminals, as Gaussian. In Section 3.4, we present a protograph LDPC code design for this channel model, and its iterative decoding threshold is compared with the one of a raptor-like LDPC code design proposed for the recently introduced fifth generation of mobile networks (5G) standard. Both code designs are compared with the Shannon limit.
- The impact of the Gaussian interference assumption on the code performance is also considered in Section 3.5. The expression of the log-likelihood ratio (LLR) and the threshold performance, computed with quantized density evolution, are presented for a single interferer when the Gaussian assumption is dropped.
- In order to get a first, yet not fully accurate, performance characteristic for the proposed LDPC codes in the random access (RA) channel, we elaborate on the decoding condition so as to abstract the physical layer in Section 4.1. A decoding region, as a function of the interference pattern for both a random code ensemble and for the LDPC code ensemble is derived. Although more accurate than the surrogate channel model, since the effective interference power and affected codeword position are considered, the abstraction is grounded on the iterative decoding threshold, and thus on large blocks assumption.
- Since RA is particularly appealing for short packet transmission, we depart from the physical layer abstraction and present in Section 4.2 physical layer simulations considering a finite block length. Interestingly, the codes designed for the surrogate channel model still perform very well on the asynchronous multiple access channel. Moreover, the performance trends and relative performance identified via the simpler simulations with the abstracted physical layer are confirmed.

## 2. System Model

#### 2.1. Asynchronous Random Access Protocol

- self-interference must be avoided, i.e., no portion of the two replicas shall overlap;
- 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;
- the delay between the start of the two replicas is drawn uniformly at random within the interval $\left(\right)$, with ${t}_{0}^{\left(u\right)}$ the activation time of User u.

- 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;
- the information about the twin location (the time position of the other replica of the same user) is extracted from the header.

**Example**

**1.**

#### 2.2. Asynchronous Random Access Channel Model

## 3. Code Design for the Asynchronous Random Access Channel

#### 3.1. Protograph LDPC Codes

#### 3.2. Code Optimization

#### 3.3. Simplified Channel Models for Code Design

- A fraction $1-\alpha $, $0\le \alpha \le 1$, of a (modulated) codeword is only affected by noise with power $2{\sigma}_{n}^{2}$.
- A fraction $\alpha $ of a (modulated) codeword is affected by noise and interference of constant power over the fraction.

#### 3.3.1. Gaussian Interference Model

#### 3.3.2. Single Interferer Model

#### 3.4. Code Design for Gaussian Interference

#### 3.4.1. Ad-Hoc LDPC Code Design

**Example**

**2.**

#### 3.4.2. Raptor-Like LDPC Code Design

#### 3.4.3. Asymptotic Results for Gaussian Interference

#### 3.5. Asymptotic Results for a Single Non-Gaussian Interferer

- Symbol-synchronous, phase-aligned, equal-power QPSK interferer, i.e., $\Delta \u03f5=0$, $\Delta \phi =0$, $\mathsf{P}=\mathsf{Z}=1$.
- Symbol-synchronous, equal-power QPSK interferer uniform at random phase, i.e., $\Delta \u03f5=0$, $\Delta \phi \sim \mathcal{U}\left(\right)open="["\; close=")">0;2\pi $, $\mathsf{P}=\mathsf{Z}=1$.

## 4. Numerical Results

#### 4.1. Abstracted Physical Layer

#### 4.1.1. Decoding Region for Random Code Ensembles

#### 4.1.2. Decoding Region for LDPC Code Ensembles

#### 4.1.3. Simulation Results

#### 4.2. Finite-Length Physical Layer Simulations with the Designed LDPC Codes

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Acronyms

5G | fifth generation of mobile networks |

ACRDA | asynchronous contention resolution diversity ALOHA |

APP | a posteriori probability |

ARA | accumulate repeat accumulate |

AWGN | additive white Gaussian noise |

CDF | cumulative distribution function |

CN | check node |

CRA | contention resolution ALOHA |

CRC | cyclic redundancy check |

CRDSA | contention resolution diversity slotted ALOHA |

CSA | coded slotted ALOHA |

DAMA | demand assigned multiple access |

DSA | diversity slotted ALOHA |

ECRA | enhanced contention resolution ALOHA |

eMBB | enhanced mobile broadband |

EXIT | extrinsic information transfer |

FEC | forward error correction |

GEO | geostationary orbit |

IC | interference cancellation |

IRCRA | irregular repetition contention resolution ALOHA |

IRSA | irregular repetition slotted ALOHA |

LDPC | low-density parity-check |

LLR | log-likelihood ratio |

LT | Luby transform |

M2M | machine-to-machine |

MAC | medium access |

MF | matched filter |

MF-TDMA | multi-frequency time division multiple access |

MRC | maximal-ratio combining |

mMTC | massive machine-type communications |

probability density function | |

PER | packet error rate |

PLR | packet loss rate |

PRACH | physical random access channel |

QPSK | quadrature amplitude shift keying |

RA | random access |

RV | random variable |

SA | slotted ALOHA |

SB | Shannon bound |

SC | selection combining |

SIC | successive interference cancellation |

SINR | signal-to-interference and noise ratio |

SNIR | signal-to-noise-plus-interference ratio |

SNR | signal-to-noise ratio |

TDMA | time division multiple access |

TS | time sharing |

VF | virtual frame |

VN | variable node |

WER | word error rate |

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**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.

**Figure 2.**Collision of multiple users (

**left**) and the abstracted surrogate model with constant interference power over $\alpha {n}_{s}$ symbols (

**right**).

**Figure 3.**The ith codeword symbol (azure) affected by a single interferer (red), with a relative epoch of $\Delta \u03f5$. Both interferer symbols $k-1$ and k impact the codeword symbol i.

**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. (

**a**) QPSK-modulated constellation after phase compensation at the receiver side. We highlight symbol ${S}_{1}=(1,0)$. If the symbol is affected by QPSK-modulated interference with the same unit power, no noise, and random phase, the received symbol will lay on the red unit circle centred in $(1,0)$. (

**b**) QPSK-modulated constellation after phase compensation at the receiver side and one QPSK-modulated interferer with the same unit power, no noise, and $\Delta \phi ={10}^{\circ}$ relative phase shift. Red crosses represent the four possible received symbols if the considered transmitted symbol was ${S}_{1}=(1,0)$.

**Figure 5.**Probability density function of the bit LLRs assuming AWGN and a symbol-synchronous QPSK interferer uniform at random phase (rnd. in the caption), or with a fixed phase offset when ${E}_{s}/{N}_{0}=6$ dB.

**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 ${\mathbf{B}}_{\mathsf{5}\mathsf{G}}$. The 5G perm.corresponds to the ensemble with base matrix ${\mathbf{B}}_{\mathsf{5}\mathsf{G}}^{\pi}$. Ad-hoc corresponds to the ensemble with base matrix ${\mathbf{B}}_{\mathsf{A}}$. Begin and end represent the iterative decoding threshold with the interference hitting the codeword from the left or the right, respectively.

**Figure 7.**The noise threshold for the code ensemble described by the base matrix ${\mathbf{B}}_{\mathsf{A}}$, 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.

**Figure 8.**PLR vs. channel load $\mathsf{G}$ for the 5G, permuted 5G and ad-hoc LDPC protograph code ensembles for the asynchronous RA setting with an abstracted physical layer. The 5G corresponds to the ensemble with base matrix ${\mathbf{B}}_{\mathsf{5}\mathsf{G}}$. The 5G perm. corresponds to the ensemble with base matrix ${\mathbf{B}}_{\mathsf{5}\mathsf{G}}^{\pi}$. Ad-hoc corresponds to the ensemble with base matrix ${\mathbf{B}}_{\mathsf{A}}$.

**Figure 9.**Dashed: PLR vs. channel load $\mathsf{G}$ for the asynchronous RA protocol with replicas protected by different $(960,480)$ LDPC codes, as a result of physical layer simulations. Solid: curves from Figure 8 assuming the abstracted physical layer. The 5G corresponds to the ensemble with base matrix ${\mathbf{B}}_{\mathsf{5}\mathsf{G}}$. The 5G perm. corresponds to the ensemble with base matrix ${\mathbf{B}}_{\mathsf{5}\mathsf{G}}^{\pi}$. Ad-hoc corresponds to the ensemble with base matrix ${\mathbf{B}}_{\mathsf{A}}$.

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## Share and Cite

**MDPI and ACS Style**

Clazzer, F.; Matuz, B.; Jayasooriya, S.; Shirvanimoghaddam, M.; Johnson, S.J.
Protograph LDPC Code Design for Asynchronous Random Access. *Algorithms* **2019**, *12*, 170.
https://doi.org/10.3390/a12080170

**AMA Style**

Clazzer F, Matuz B, Jayasooriya S, Shirvanimoghaddam M, Johnson SJ.
Protograph LDPC Code Design for Asynchronous Random Access. *Algorithms*. 2019; 12(8):170.
https://doi.org/10.3390/a12080170

**Chicago/Turabian Style**

Clazzer, Federico, Balázs Matuz, Sachini Jayasooriya, Mahyar Shirvanimoghaddam, and Sarah J. Johnson.
2019. "Protograph LDPC Code Design for Asynchronous Random Access" *Algorithms* 12, no. 8: 170.
https://doi.org/10.3390/a12080170