## 1 Introduction  
Continuous phase modulation (CPM) is a family of nonlinear modulation schemes with phase continuity and constant envelope. It has been widely used in mobile communication [1], aeronautical telemetry standard [2], industrial communications standards [3], and future satellite broadcasting [4]. It has the advantages of a high-frequency spectrum and power efficiency, as well as a high power of resistance to channel nonlinearity. On the other hand, CPM suffers from the high implementation complexity of synchronization and symbol detection. Multi-_h_ CPM, developed with more than one modulation index \(h\), has a higher power and spectral efficiency than single-_h_ CPM and provides three times the spectral efficiency of PCM/FM [5, 6]. Multi-_h_ CPM has been selected as the Tier II waveform of the US Advanced Range Telemetry Program Organization (ARTM) [7]. The multiple modulation indices change periodically, which can be coded to improve spectrum efficiency further. Thus, it promises broader application scenarios than the single-_h_ CPM. However, more modulation indices bring higher complexity of the trellis and more matched filters, leading to an extremely high detection complexity. Meanwhile, synchronization also has high implementation complexity since the decision-directed algorithm is generally used for the coherent receiver. The maximum-likelihood sequence detection (MLSD) is used to obtain optimal detection performance for CPM signals. Some decoding algorithms are commonly used to implement the MLSD according to the trellis in the receiver, such as the Viterbi algorithm and the BCJR algorithm [8; 9]. It provides significant detection gain compared with the symbol-by-symbol method. However, tremendous complexity is introduced by the trellis of a large number of phase states and branches. Due to its high complexity, some reduced-complexity methods are proposed, such as tilted phase transformation (TPT) [10; 11], frequency pulse truncation (FPT) [11; 12], pulse amplitude modulation (PAM) [13; 14], and state-space partitioning (SSP) [11; 15]. All the above methods have been discussed in [16], and have been widely used in various CPM receivers [11; 17; 18; 19; 20]. The above algorithms may cause some performance loss in demodulation, except for the TPT method. In practical applications, combining several of them brings a better tradeoff between implementation complexity and performance. These methods also result in a reduction in synchronization complexity. In [21], the decision-directed synchronization for joint phase and timing recovery is introduced with ML estimation for phase and timing error. The synchronization with FPT is also proposed in [22]. The Walsh signal space and PAM decompositions help reduce the synchronization complexity in [23; 24; 25; 26], respectively. In the conventional joint carrier and timing recovery methods mentioned above, the ML estimation of timing error is always calculated by the derivative matched filters because of the nonlinear function of timing offset in the log-likelihood function. It is commonly approached by a finite difference of the outputs from two matched filter banks, of which delays of the on-time MF banks are early and late, respectively, represented as the early-late (EL) synchronizer [27; 28]. The EL-based synchronization algorithm is widely utilized for linear [29; 30; 31] and nonlinear [21; 22; 26; 32] modulation schemes, since it offers high estimation accuracy with low computational cost. The general methods to reduce the complexity of the synchronization depend mainly on the simplification methods of MLSD. In this paper, we derive a low-complexity synchronization for the multi-\(h\) CPM. We combine the reduced-complexity MLSD methods of LPA, TPT, and SSP to reduce the number of phase states in the detection trellis as conventional methods. Additionally, we pursue reducing the synchronization complexity of the error signal estimator based on the LPA to the phase of the multi-\(h\) CPM. With LPA, the timing delay is linearized to the phase.