On the Application of the CramerRao and Detection, Theory Bounds to Mean Square Error of, Symbol Timing Recovery

The Cramer-Rao bound is often used to lower bound the mean square error of symbol timing estimators. However, the valid application of this bound requires that the estimator be unbiased and that the signaling waveform be sufficiently smooth. In some cases of practical interest, these conditions are not satisfied. Furthermore, the Cramer-Rao bound is known to yield poor results for small signal-to-noise ratios. A modified Cramer-Rao bound is derived that takes account of the fact that the symbol timing estimate is restricted to a finite interval of one symbol duration. A detection theory bound is also applied to the symbol timing problem. Results obtained using the Cramer-Rao and detection theory bounds are compared. It is shown that the detection theory bound is superior to the traditionally used Cramer-Rao bound. In particular, the detection theory bound gives useful results for sharp pulse shapes and shows a dependence of the mean square error on the data sequence. It also yields meaningful results for small signal-to-noise ratios.