In this paper, we derive a joint maximum likelihood (ML) algorithm for the estimation of frequency offset, carrier phase and symbol timing for continuous phase modulations (CPMs). We have considered a burst-mode scenario over additive white Gaussian noise (AWGN) channels in which an optimized training sequence is embedded within each burst in order to assist the synchronization task. The proposed data-aided (DA) approach takes advantage of the optimum training sequence structure which can be applied to the entire CPM family. The simulation results show that the estimator performs quite close to the theoretical Cramér-Rao bound (CRB) for all synchronization parameters in terms of their estimation error variances even at low signal-to-noise ratios (SNRs).

In this paper, we study the Cramer-Rao bound (CRB) for continuous phase modulation (CPM) signals where frequency offset, carrier phase, and symbol timing are jointly estimated when transmitted over an additive white Gaussian noise (AWGN) channel. We consider a data-aided (DA) estimation scenario in which the estimator takes advantage of a known training sequence at the start of each burst. Thus, we first derive the joint CRBs as functions of a known training sequence and CPM parameters. By analyzing the CRB expressions, we propose the optimum training sequence for which the CRB is minimized. We show that the same training sequence is optimum for all three estimation parameters. Additionally, we compare the performance of the optimum training sequence with a random one by providing a closed-form expression for the unconditional CRB (UCRB) for symbol timing estimation of CPM signals. Comparing the UCRB and the CRB for the optimum training sequence reveals that a DA estimator with the optimum training sequence leads to significant gains in terms of the mean-square error of the estimation parameter when the underlying CPM scheme is non-binary and/or partial response.

In this paper, we propose a complete synchronization algorithm for continuous phase modulation (CPM) signals in burst-mode transmission over additive white Gaussian noise (AWGN) channels. The timing and carrier recovery are performed through a data-aided (DA) maximum likelihood algorithm, which jointly estimates symbol timing, carrier phase, and frequency offsets based on an optimized synchronization preamble. Our algorithm estimates the frequency offset via a one-dimensional grid search, after which symbol timing and carrier phase are computed via simple closed-form expressions. The mean-square error (MSE) of the algorithm's estimates reveals that it performs very close to the theoretical Cramer-Rao bound (CRB) for various CPMs at signal-to-noise ratios (SNRs) as low as 0 dB. Furthermore, we present a frame synchronization algorithm that detects the arrival of bursts and estimates the start-of-signal. We simulate the performance of the frame synchronization algorithm along with the timing and carrier recovery algorithm. The bit error rate results demonstrate near ideal synchronization performance for low SNRs and short preambles.

In this paper, we study the Cramér-Rao bound (CRB) for continuous phase modulation (CPM) signals. Based on the CRB computations, we propose the optimum training sequence for joint estimation of carrier phase, frequency offset and symbol timing. Our proposed training sequence is optimum in the sense that it minimizes the CRBs for all three estimation parameters simultaneously. The results are useful in the design of training sequences attached to data packets in burst-mode transmissions where data-aided (DA) synchronization is employed.