This repository contains the matlab numerical routines of the paper:

[1] K.-H. Ngo, A. Lancho, G. Durisi and A. Graell i Amat, "Unsourced Multiple Access With Random User Activity," submitted to IEEE Trans. Inf. Theory, Jan. 2022.

This work was presented in part in

[2] K.-H. Ngo, A. Lancho, G. Durisi and A. Graell i Amat, "Massive Uncoordinated Access With Random User Activity," in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Melbourne, Australia, Jul. 2021, pp. 3014-3019.

Please, cite the aforementioned papers if you use this code.

This repository contains the codes to evaluate random-coding bounds on the error probability of an unsourced multiple access (UMA) system, and the required energy per bit (EbN0) for such probability to lie below a certain threshold. There are four folders:

1. /RCU_KaKnown/: RCU bound on the per-user probability of error (PUPE) for an UMA system where the number of active users, denoted by Ka, is known.

• RCU_KaFixedKnown.m: RCU bound for the case where Ka is fixed and known. This is a straightforward extension of the bound in the following seminal paper to the complex-valued case:

[3] Y. Polyanskiy, "A perspective on massive random-access," in Proc. IEEE Int. Symp. Inf. Theory (ISIT), 2017, Aachen, Germany, pp. 2523-2527.

To evaluate the RCU bound in [3], simply adjust the framelength, i.e., the degrees of freedom.

• RCU_KaRandomKnown.m: RCU bound for the case where Ka is random (follows a distribution with specified PMF) and known. This bound is obtained by averaging the bound above for fixed and known Ka over the distribution of Ka.
• RCU_KaPoissonKnown.m: RCU bound for the case where Ka follows a Poisson distribution and is known.
• EbN0_KaPoissonKnown.m: Find the minimal required EbN0 such that the RCU bound for Ka following a Poission distribution and known lie below a certain threshold.
• binary_search.m, golden_search.m: Auxiliary functions.

2. /RCU_KaUnknown/: RCU bounds proposed in [1], [2] on the misdetection (MD) and false alarm (FA) probabilities for an UMA system where Ka is random and unknown.

• RCU_KaRandomUnknown.m: RCU bounds for the case where Ka follows a distribution with specified PMF ([1, Theorem 1]).
• RCU_KaPoissonUnknown.m: RCU bounds for the case where Ka follows a Poisson distribution.
• RCU_floor_KaRandomUnknown.m: Error floors characterized in [1, Theorem 3].
• EbN0_KaPoissonUnknown.m: Find the minimal required EbN0 such that the RCU bounds on the MD and FA probabilities lie below certain thresholds.
• binary_search_P_MDFA.m, golden_search_P1_MDFA: Auxiliary functions.

3. /SA-MPR/: Application of the bounds above for slotted ALOHA (SA) with multi-packet reception (MPR).

• EbN0_SAMPR_KaPoissonKnown.m: Find the minimal required EbN0 for SA-MPR, Ka follows a Poisson distribution and is known.
• EbN0_SAMPR_KaPoissonUnknown.m: Find the minimal required EbN0 for SA-MPR, Ka follows a Poisson distribution and is unknown. See [1, Corollary 2].

4. /TIN/: RCUs bound for a scheme that treats interference as noise (TIN).

• EbN0_TIN_KaPoissonUnknown.m: Find the minimal required EbN0 for TIN, Ka follows a Poisson distribution and is unknown.

Note: the codes for the SPARC and enhance SPARC schemes are provided by the group of Krishna R. Narayanan, and hence not included in this repository.