Low-Complexity Channel and Carrier Frequency Offset Estimation for Burst-Mode CPM Using Optimized Training Waveforms

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.