# A Carrier-Based Gardner Timing Synchronization Algorithm for BPSK Signal in Maritime Communication
## 1 Introduction  
Medium-wave communication is a radio communication technique that uses electromagnetic waves with a wavelength of 1000 m\(\sim\)100 m (a frequency range of 300-3000 kHz) and is an important component of maritime communication. Medium-wave communication is widely used for maritime emergency rescue, aircraft and ship communication, and radio navigation. However, in digital communication systems, the received carrier signal is affected by the Doppler effect and the local crystal frequency offset, which creates frequency offset in the frequency domain and changes the length of the signal in the time domain. To synchronize the carrier signal with the local clock signal, carrier synchronization and bit synchronization are required at the receiver. Bit synchronization, also known as symbol synchronization, extracts the timing pulse sequence for sampling judgments at the receiver side and is the foundation for accurate sampling judgments. Bit-synchronization algorithms can be divided into feedforward synchronization algorithms and feedback synchronization algorithms according to the algorithm structure. It can also be classified as data-assisted and non-data-assisted algorithms [1; 2]. The classical synchronization algorithms include the early-late gate (E-LG) algorithm [3], M&M (Mueller & Muller) algorithm [4], maximum likelihood algorithm [5], wave division multiplex (WDM) algorithm [6], and Gardner algorithm [7]. Among them, refs. [3; 4; 6; 7] belong to timing recovery synchronization algorithms. The Gardner timing recovery bit synchronizationalgorithm requires only two resamples in one symbol for error detection. The carrier phase does not affect the synchronization judgment performance, data assistance and judgment feedback are not required, and it is easy to implement in high-speed digital communication systems. Hence, it is widely used in engineering practice. Gardner algorithm is continuously applied to quadrature amplitude modulation (QAM) signals [8; 9; 10], multiple phase shift keying (MPSK) signals [11; 12], Gaussian minimum frequency shift keying (GMSK) signals [13], and other modulation systems [14; 15; 16], with an ever-expanding range of applications. The Gardner algorithm works by resampling the rough demodulated baseband signal. The feasibility of the algorithm depends on an assumption that the phase deviation of each sampled value is constant, which means there is no carrier frequency deviation. When the assumptions are not satisfied, the performance of the algorithm may degrade. As a result, improved algorithms for correcting frequency deviation are proposed [10]. In addition, Gardner algorithm introduces a certain amount of self-source noise when the roll-off factor (ROF) is small and adjacent symbols are the same, which affects the performance of bit synchronization. Improved algorithms using pre-filtering method [8; 17] and compensation method [18; 19] have been gradually proposed for Gardner algorithm with large self-jitter in systems with small ROF. However, the complexity of these improved algorithms is relatively high. For the method applicable to MPSK modulation proposed in [11], ref. [20] proposed an improved algorithm applicable to higher-order MPSK modulation. Ref. [21] improved the Gardner phase detector (GPD) so that the algorithm still performs well in QAM modulated systems with the ROF close to 0. When adjacent symbols are the same, the symbol difference is always zero and the Gardner algorithm cannot capture the timing error, which makes the Gardner algorithm generate large timing jitter [22] and leads to a high level of self-noise in the system. Some improved algorithms were proposed to address this issue [23]. Correction algorithms [9] and preprocessing algorithms [24] were proposed to mitigate self-noise in the Gardner algorithm. To solve the problem of long synchronization timing time, an improved method that increases the gain control processing was proposed in [25].