## 1 Introduction
In this study, we delve into the computational process of the MSD algorithm, encompassing two distinct stages: correlation calculation and symbol decision. The correlation calculation necessitates substantial complex multiplication and addition operations, constituting the major computational load in the entire MSD algorithm. To alleviate the resource overhead, the SC is utilized upon conducting correlation calculations due to its characteristic simple computing circuit. Moreover, the correlation calculation requires summing a series of data. Given the constraint in SC that the result must fall within the interval of 0 to 1, this study introduces a stochastic adder with an adaptable scaling factor to guarantee that the summation results meet the specified range. Furthermore, considering the need to identify the maximum value among all correlation results for the subsequent symbol decision, selecting an appropriate scaling factor in stochastic addition also ensures discrimination between different computational results, thereby guaranteeing the detection performance of the entire MSD algorithm. When it comes to symbol decision, the modulus of the correlation results must be calculated. If the SC is adopted in this process, a new random sequence should be generated based on the corresponding results. However, the complete independence of the regenerated random sequence from the original sequence cannot be guaranteed. This will result in decreased accuracy of the computational results, thereby leading to a loss in the detection performance of the MSD algorithm. Considering the limitations of SC, binary computing is adopted for the symbol decision in the MSD algorithm. Additionally, the errors in binary computing are due to minor deviations from data truncation during the bit quantization process. Therefore, the adoption of binary computing becomes a viable option in the process of symbol decision. Due to the requirement of a certain computation time to ensure the convergence of results in SC-based correlation calculation, there is ample computation time for the binary-computing-based symbol decision. Through meticulously designing a pipeline architecture, the hardware resources needed for binary computing can be significantly reduced, thereby optimizing the overall computational efficiency. This study makes the follow contributions:  
* A hybrid stochastic computing architecture for the MSD algorithm is proposed. By utilizing SC and binary computing at various computational stages in the MSD algorithm, based on the intrinsic properties of the different computing paradigms, the proposed method significantly reduces hardware resource consumption while maintaining the signal detection performance. * Considering the constraints on the range of data values in SC, a scalable stochastic adder is proposed to achieve the summation of different sequences. This adder can permit different scaling factors, thereby facilitating the selection of optimal parameters tailored to various input conditions. This ensures that the computational results do not exceed the prescribed limits, while simultaneously guaranteeing computational accuracy. * Due to the low update rate of the SC results, a pipeline architecture is proposed to maximize the utilization of computing time for the symbol decision of the MSD algorithm. The fully serial computation approach is employed to minimize the resource overhead in binary computing. The rest of this paper is organized as follows. Section 2 introduces the related works, including those on the MSD algorithm, FM signal demodulator, and the fundamental computational methods of SC. Section 3 describes the hardware circuits of the basic SC units and presents an architecture for the proposed hybrid computing scheme that integrates stochastic and binary computing. Section 4 analyzes the computational accuracy and hardware resource consumption of the basic SC unit. Section 5 assesses the computational accuracy, BER performance, and hardware resource of the MSD algorithm based on hybrid SC.