We present algorithms of the channel and carrier frequency offset (CFO) estimation for burst-mode continuous phase modulation (CPM) transmission with optimized training waveforms, which are obtained through minimizing the Cramer-Rao bound (CRB). Wirtinger calculus, which provides a framework for complex-valued signal processing, is used to simplify the derivations of CRBs and maximum likelihood estimation. Unlike linear modulation, the training waveform is optimized not only with respect to the modulated training sequence but also the sampling delay and sampling rate due to the continuous phase of CPM waveforms. In order to reduce the computation complexity of the optimization, we derive a closed-form expression for efficient computation of the objective function. The numerical results demonstrate the significant performance improvement of the optimized waveforms compared to the training sequences of previous works. Based on that, we propose two schemes for CFO estimation. The first scheme is a suboptimal maximum likelihood estimation of the CFO and it achieves a better trade-off between the estimation accuracy and complexity compared to optimal maximum likelihood estimation. The second scheme is based on the channel estimation, which leads to much lower implementation complexity than the other estimators. Finally, the measured bit error rate reveals the considerable gain of the estimation algorithm with the optimized training waveform.

In this study, they authors introduce a suboptimal training sequence for burst‐mode continuous phase modulation. The optimum training sequence has been prove to be optimal for the estimates of carrier frequency offset, carrier phase error and timing error. However, the optimum sequence brings high side‐lobe of the log‐likelihood function (LLF), that is not benefit for detecting the start of the sequence (SOS). Thus, the suboptimal sequence is designed to improve the SOS performance through decreasing the side‐log effect of the LLF. Meanwhile, the proposed sequence can minimise the Cramer‐Rao bound for the joint estimation of carrier frequency offset, carrier phase error and symbol timing error. The data‐aided maximum likelihood algorithms for timing error and phase error estimation are derived for evaluating the performance of the proposed sequence. The simulation results demonstrate that the new sequence provides better performance on start detection of the sequence than the optimal one. The performance of timing and carrier phase recovery is also close to the ideal synchronisation performance.