Skip to Main Content
The detection of pulse amplitude modulation (PAM) carrier signals requires accurate symbol timing and carrier phase references. In most cases, it is desired to estimate these parameters directly from measurements on the received data signal. This paper adds to and unifies the theory of maximum likelihood [ML] estimation as applied to PAM timing and phase recovery. Several different estimation strategies are considered. Data-aided [DA] estimators are found which assume the transmitted data symbols are known at the receiver. Nondata-aided [NDA] estimators are found which require only knowledge of the statistics of the transmitted data symbols. Structures for estimation of symbol timing, carrier phase, and joint estimation of timing and phase are presented. The estimators are evaluated on the basis of their error variances. Relatively simple approximate expressions for these error variances are presented. These expressions allow the comparison of the effects of excess bandwidth, different modulation schemes, DA versus NDA recovery, and joint estimation versus estimation of only one parameter. A practical implementation of the ML estimator, termed a pseudo-maximum likelihood (PML) estimator, is proposed and analyzed. The performance of the PML estimator is shown to include a noise-independent, data-dependent jitter which dominates in many cases of practical interest.