Design and analysis of feedforward symbol timing estimators based on the conditional maximum likelihood principle

This paper presents a general feedforward symbol-timing estimation framework based on the conditional maximum likelihood principle. The proposed timing estimator presents reduced implementation complexity and is obtained by performing an approximation on the Fourier series expansion of the conditional maximum likelihood function. The proposed algorithm is applied to linear modulations and two commonly used continuous phase modulations: minimum shift keying (MSK) and Gaussian MSK (GMSK). For the linear modulations, it is shown both analytically and via simulations that the performance of the proposed estimator is very close to the conditional CRB and modified CRB for signal-to-noise ratios (SNRs) in the range SNR ≤ 30 dB. Furthermore, the proposed estimator is shown to be asymptotically equivalent to the classic square-law nonlinearity estimator under certain conditions. In the case of MSK and GMSK modulations, although the proposed algorithm reaches the conditional CRB at certain SNRs, however, the conditional CRB is quite far away from the modified CRB, and there exists an alternative algorithm whose performance comes closer to the modified CRB. Therefore, the proposed estimator is more suitable for linear modulations than for MSK and GMSK modulations.