# Comparative Analytical Study of SCMA Detection Methods for PA Nonlinearity Mitigation
## 1 Introduction  
Next-generation communication systems must be capable of providing several users/devices with appropriate service levels for the industrial internet of things (IIoT) and Industry 4.0 [1]. In the context of multiple-access techniques for these ecosystems, non-orthogonal multiple access (NOMA) has emerged as a promising solution that has the potential to support several users over a finite number of temporal/spectral resources. NOMA-based approaches are broadly categorized into the following types [1, 2]: (a) power domain NOMA (PD-NOMA), and (b) code domain NOMA. PD-NOMA uses superposition coding to overlap multiple users and detects corresponding user symbols on the receiver side by successive interference cancellation (SIC) or message passing algorithms (MPAs). However, PD-NOMA is known to support a limited number of users due to inter-layer error propagation, and its reliance on power diversity [3, 4, 5]. Apart from PD-NOMA, specific code-domain NOMA-based approaches, like sparse code multiple access (SCMA) have recently been found to be particularly promising [6, 7, 8, 9], as they not only allow for potential coding/shaping gains through codebook design, but also enable near-optimal detection using MPAs. Besides, SCMA is also known for its robustness to error propagation.  
However, transmit-side power amplifier (PA) nonlinearities have been found to degrade the performance of generic SCMA systems. From [PERSON]'s theorem [10], transmit-side PA nonlinearity is known to add an independent equivalent distortion noise term that lowers the overall signal-to-noise ratio. Two types of competing MPA-based detection methods exist to mitigate this degradation: (a) Bussgang decomposition-based MPA detectors [11] and (b) random Fourier feature (RFF)-based detectors [12]. While decomposition-based approaches achieve commendable performance under a limited implementationbudget, the RFF based approaches offer benefits like universal approximation and generalization across various types of nonlinear PA characteristics. However, RFF-based approaches have slightly more computational overhead, and in certain hardware limited IIoT ecosystems, the implementation complexity of algorithms outweighs the error-floor reached subject to the achievement of a minimum level of quality of service (QoS) [13; 14; 15]. Therefore, it is compelling to compare and derive analytical insights/comparisons on the error floors of the Bussgang-based MPA methods and to decide on the suitability of a detector for a given bit error rate (BER)-based on the QoS. Several works in the literature have studied the nonlinearity effect not only in SCMA but also in other environments, such as [16], where a Bussgang-based receiver design was proposed for nonlinear PD-NOMA. Moreover, in [12], a nonlinear SCMA system model was studied, and a RFF-based solution was proposed to improve BER performance as equivalent to that obtained in the presence of a linear AWGN channel, whereas an iterative method based on clipping noise was proposed in [11]. Additionally, in [17], RFF-KLMS based algorithm was proposed to mitigate nonlinearity in MIMO-VLC channels.  
_Contributions_: In this paper, we present rigorous analytical studies and insights on the optimality of the Bussgang-based MPA for downlink SCMA with PA impairments. From our analysis, the Bussgang-based MPA detector is found to reach a non-negligible BER floor compared to the universally optimal RFF-based MPA, and the analytical results are presented to quantify the BER floor. Next, these results are validated using computer simulations under different fading distributions. The quantification of this error floor could potentially allow for switching between detection methods in hardware-constrained IIoT environments, where meeting a specific QoS constraint with minimal computations is of paramount importance.