## 1 Introduction  
Target range telemetry plays a significant role in the aerospace field. Continuous phase modulation (CPM) signals, characterized by a constant envelope and continuous phase, achieve high spectral and energy efficiency, rendering them widely used in target range telemetry [1, 2]. As a typical form of CPM signals, pulse code modulation/frequency modulation (PCM/FM) has long been the primary communication scheme in telemetry systems due to their ability to resist flame and phase interference [3; 4; 5]. However, as the transmission distance increases and the volume of telemetry data grows, the limitations of traditional detection methods are becoming increasingly apparent [6]. Maximum likelihood sequence detection (MLSD) theoretically achieves optimal demodulation performance; however, its high computational complexity and substantial challenges in practical implementation limit its applicability. Many researchers have proposed methods such as pulse truncation [7] and state-space partitioning [8] to reduce the computational complexity of the algorithm, yet the complexity remains a critical issue. Multi-symbol detection (MSD) algorithms have been proposed and have gained increasing attention to further balance demodulation performance and algorithmic complexity. This algorithm leverages the continuous phase variation characteristics of symbols to perform joint detection of multiple symbols, thereby enhancing demodulation. [PERSON] et al. first proposed the MSD algorithm in 1971 for the demodulation of frequency shift keying (FSK) signals, achieving superior performance in non-coherent demodulation conditions [9]. In 1974, [PERSON] et al. applied the MSD algorithm to demodulate continuous phase frequency shift keying (CPFSK) signals, further improving CPFSK detection performance [10]. In 2000, [PERSON] et al. extended the MSD algorithm to the detection of PCM/FM signals, achieving a 2.5 dB gain compared to traditional single-symbol detectors [11]. However, these MSD algorithms rely on correlating the received signals within the observation interval with all possible transmitted signal waveforms (local reference signals), resulting in exponential growth in complexity as the observation interval increases [12]. To reduce the computational complexity of MSD algorithms while maintaining their superior detection performance, numerous improved MSD algorithms have been proposed. The Baseband Quadrature Complex Rotation (BQCR) MSD algorithm significantly reduces the computational burden of correlation computations through complex rotation operations [13]. However, its reliance on a high number of signal sampling points makes it more suitable for scenarios with low baseband rates, presenting certain limitations [14]. [PERSON] et al. classified local reference signals into two categories based on whether the intermediate symbol is \"0\" or \"1\", and by performing correlation operations only with one category, they then achieved signal demodulation by comparing the threshold with the correlation peak. This method reduced the computational complexity of the MSD algorithm by 50%. Nonetheless, the demodulation performance of this method is highly dependent on the precise setting of the threshold and is sensitive to noise, which limits its application [15]. Similarly, [PERSON] et al. proposed using already-decoded symbols to filter local reference signals, which reduced the computational complexity by 50%. However, when errors occur in the decision of earlier bits, it is prone to error propagation, leading to a loss in demodulation performance [16]. However, the above MSD algorithms still face high computational complexity and difficult hardware deployment when applied to receivers that need to handle multiple signals simultaneously. This paper proposes a low-complexity multi-symbol detection algorithm based on average-matched filtering and phase symmetry. Then, the simplified MSD algorithm is implemented on the FPGA platform through time-sharing multiplexing filters to save hardware resources.