## 1 Introduction  
The advent of the fifth and beyond fifth generation (5G) of wireless communication networks brings along with it the massive connectivity of users and devices, short message packets, low latency, sporadic communication, and low power, just to mention a few aspects. One of the frontrunners among the technologies that make all of these possibilities a reality is the non-orthogonal multiple access (NOMA) scheme for multiuser transmission in networks. The NOMA scheme has become the transmission scheme of choice due to some downsides associated with its orthogonal multiple access (OMA) scheme counterparts, which were used in previous generations of wireless communication networks. Some of these downsides include restriction to the number of users that OMA's available resources can simultaneously cater for, and the use of dedicated feedback path-based scheduling techniques, resulting in additional overhead in terms of signally for orthogonality maintenance, just to mention a few. On the other hand, the NOMA-based 5G wireless communication network [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] is capable of supporting multiple users' connectivity via non-orthogonal resources. One of the major signal processing tasks that is required for optimal performance of NOMA-based 5G wireless networks is the detection of multiple signals with the aid of a multiuser detector (MUD). This detection is more challenging in the uplink grant-free (GF) transmission mode employed in NOMA systems. Usually, GF-free access is considered in uplink in a bid to reduce signaling overhead in the system. In the uplink GF-NOMAsystem, prior knowledge of user activity is not available at the base station (BS), because each active user sends their information data without needing to obtain permission via an uplink grant request. Consequently, the BS must implement detection of both multiuser activity and user signals at the same time. There have been different approaches and algorithms developed to implement multiuser detection in uplink GF-NOMA wireless communication networks in the literature. Some of these are presented below. The first category is those techniques that employ multiple measurement vector (MMV) models in the design of the MUD. In [11], the authors develop a simultaneous orthogonal matching pursuit (SOMP)-based MUD to recover the transmitted signal by converting the single measurement vector (SMV) model into an MMV model. The authors in [12] adopt a differential OMP algorithm to develop a MUD scheme for a system with an MMV model. In [13], a compressive MUSIC-based MUD is constructed on a compressive MUSIC algorithm, a deterministic extension of compressive sensing (CS), to achieve detection of transmitted signals in the uplink NOMA system. A joint sparsity-based greedy algorithm recovery is proposed in [14] for MMV problems, named SOMP-EXT, which operates based on the mechanism of SOMP. The main function of this algorithm is to estimate the ratios of a posteriori probability for each of the sparse signals and return the extrinsic information via an iterative mechanism. One of the common drawbacks of the aforementioned methods is the huge associated computational complexity involved as a result of the multiple measurement vector-based models that are used, with no significant performance justifying the huge computational cost. Hence, not many works have been documented in this area. The second category includes techniques that employ the SMV model. Among the works in this category is that presented in [15], in which a compressive sensing algorithm without knowledge of the sparsity level, as proposed in [14], is extended for the estimation of multiuser activity and the detection of users' signals in uplink GF-NOMA systems. Another is the method presented in [16], in which an iterative order recursive least square (IORLS) algorithm that uses sparsity for each frame is proposed. The IORLS recursively uses a modified orthogonal matching pursuit (OMP) mechanism within a frame to extract the sparsity support information.