Analysis of two loops for carrier recovery in CPM with index 1/2

An examination is made of the performance of a squaring loop and a fourth-power loop used for tracking the received carrier phase of binary continuous-phase modulated (CPM) signals with index 1/2. Noise-free analysis is performed first and a simple relationship between the phase error and the control voltage is obtained for each case, with the premodulation filter response as a parameter. Linearized analysis of the phase error variance is then conducted in the presence of additive white Gaussian noise. It is shown that the phase variance in both cases contains a self-noise component that sets a lower bound on the synchronizer´s performance at very high signal-to-noise ratios. Numerical results are presented for a number of popular CPM signals. It is found that the squaring loop performs best with MSK (minimum shift keying), while the fourth-power loop performs best with duobinary MSK