## 1 Introduction  
Considering the global seamless coverage of 6G networks, high-speed aircraft require stable wireless communication connections to guarantee the effective execution of diverse missions [1, 2]. Due to its insensitivity to large Doppler shifts, Frequency Modulation (FM) technology has been widely used in aircraft communications standards [3] such as IRIG-106 [4] and DVB-S2 [5]. To ensure the quality of signal demodulation, the Multiple Symbol Detection (MSD) algorithm [6] is a key signal processing technique used in the FM receiver. Figure 1 shows the bit error rate (BER) performance of the three-symbol MSD algorithm under different Doppler shifts, from which it can be seen that, when the symbol rate is 10 Msps and the Doppler shift is 100 kHz, the BER loss is almost negligible. Therefore, FM technology exhibits remarkable superiority in resisting Doppler shifts. The MSD algorithm can enhance the signal detection performance by leveraging the continuity between different symbols based on the maximum likelihood principle. The essence of the MSD algorithm lies in its ability to holistically process multiple consecutively received symbols as a complete unit, transcending the limitation of relying solely on a single acquired symbol [7]. MSD algorithms can overcome the BER loss of traditional single-symbol detection algorithms in low signal-to-noise ratio (SNR) scenarios, effectively reducing the demodulation threshold of the receiver [8; 9]. However, due to the need to calculate correlation results between the received signal and all possible local signals, a large number of multiplication and addition operations are required, leading to considerable computational complexity [10]. Therefore, this study starts from fundamental computational units such as multipliers and adders, and leverages stochastic computing (SC) to mitigate the computational complexity of MSD algorithms. As a novel computational paradigm, SC has been proposed to effectively reduce the complexity of hardware circuits and improve their computational efficiency [11; 12; 13], and has been applied to a wide variety of applications such as neural networks [14; 15; 16], image processing [17; 18; 19], and channel coding [20; 21; 22]. SC uses random bits to encode binary numbers [23]. Information is contained in a series of random bits, where the probability of logical 1 represents the target values. In SC, as the computational process is transformed into operations on single-bit data streams, arithmetic circuits exhibit exceptional simplicity [24]. Specifically, for unipolar input data, the multiplier circuit is an AND gate, and for bipolar input data, the multiplier is an XNOR gate. This simple structure presents significant advantages over alternative approximation paradigms, such as fixed-point arithmetic [25; 26] or neural-based approximations [27; 28; 29]. However, the correlation between two stochastic sequences in the SC context has a pivotal influence on the computational outcomes [30]. This correlation can potentially necessitate a longer sequence to ensure the reliability of the computational results. Consequently, this may compromise the real-time performance of the system, posing challenges in terms of timeliness and responsiveness. Due to its ability to enhance the computational efficiency of SC, the hybrid SC has attracted the attention of researchers [31; 32; 33; 34; 35]. SC exhibits low resource consumption, yet demonstrates inferior computational accuracy compared to binary computing. Conversely, binary computing achieves higher computational precision at the expense of higher computational complexity compared to SC. Hybrid computation can leverage the unique advantages of both stochastic and binary computing, while simultaneously addressing their respective limitations. Through integrating these two computing paradigms, hybrid  
Figure 1: The BER of 3-symbol MSD algorithm under different Doppler shifts. computation offers a comprehensive solution that maximizes the benefits and minimizes the drawbacks of each individual method.