Zero-Crossing timing error detection for estimating and compensating timing error.

The Zero-Crossing timing error detector is a decision directed timing error detector similar to the (modified) Mueller-Muller algorithm. It is based on the zero-crossing of the input signal $x(n)$. In contrast to Gardner and Early-Late, the signal does not need to be two times oversampled. However, it strongly depends on the carrier phase. This means it needs to be applied after phase recovery. Otherwise the estimates are prone to error.

Due to its decision directed nature it is dependent on knowledge about the used constellation. A hard decision is performed on the input signal $x(n)$ to get symobl estimate $s_n$. $$ s_n = \mathrm{argmin}_k |x(n) - c_k|^2 $$ Here, $c_k$ refers to the constellation points (e.g. 16-QAM).

The equations for estimating the error in the modified Mueller-Muller algorithm for a given input signal $x(n)$ is given by $$ t_\mathrm{err} = \mathrm{Re}\left\{\sum_{n=0}^{N-1}\left(s_{n+1} - x(n-1)\right)x^*(n)\right\}. $$ $N$ again refers to the averaging length.