Irregular Repetition Slotted ALOHA Over Rayleigh Block Fading Channels: Bounds and Threshold Saturation via Spatial Coupling

Abstract

In irregular repetition slotted ALOHA (IR-SA) systems, a population of devices transmits their packets to an access point (AP) within a frame of slots. The AP decodes these packets by iterative processing between intra-and inter-slot successive interference cancellations. The average normalized offered traffic, as a performance metric, represents the number of packets transmitted per slot when the packet loss rate approaches zero. Such asymptotic types of traffic as the belief propagation (BP) threshold, the maximum a posteriori (MAP) threshold, and the converse bound of IR-SA systems have been analyzed over various channel models. However, over fading channels, the MAP threshold and the converse bound have not yet been investigated. This paper derives an MAP threshold and a converse bound of the systems over Rayleigh block fading channels. The derivations are based on two extrinsic information transfer (EXIT) curves, which are associated with two iterative density evolution equations to analyze the BP threshold of the IR-SA systems. First, since an open decoding tunnel exists in an EXIT chart, the sum of the two areas below two EXIT curves is smaller than the area of the entire domain. This provides the traffic's converse bound, which is tight. Second, a coincidence of the BP EXIT and MAP EXIT curves makes it possible to derive the traffic's MAP threshold. Third, a density evolution for a spatially-coupled scheme is formulated and gives a BP decoding threshold of the traffic. Numerical results show that the spatially-coupled scheme achieves a threshold saturation effect where the BP threshold approaches the MAP threshold.INDEX TERMS belief propagation (BP) threshold, extrinsic information transfer (EXIT) curve, maximum a posteriori (MAP) threshold, slotted ALOHA, spatial coupling