Lower Bounds for the Performance of Iterative Timing Recovery at low SNR

The push for higher recording densities has motivated the development of iterative errorcontrol codes of unprecedented power, whose large coding gains enable low error rates at very low SNR [1] [2]. In addition, the iterative decoding technique has been extended to turbo equalization, where the equalizer and the decoder iterate [3]. Consequently, timing recovery, which typically derives no benefit from coding, must be performed at an SNR lower than ever before. At high SNR, the timing recovery process can be separated from the decoding process with little penalty; timing recovery can use an instantaneous decision device to provide tentative decisions that are adequately reliable, which can then be used to estimate the timing error. In essence, the timing recovery process is able to ignore the presence of the code, and assume instead that neighboring symbols are independent. At low SNR, however, timing recovery and decoding are intertwined. The timing recovery process must exploit the presence of the code to get reliable decisions, and the decoder must be fed well-timed samples to function properly. In principle one could formulate the problem of jointly determining the maximum-likelihood (ML) estimates of the timing offsets and message bits, but the complexity would be prohibitive. A solution based on the expectation-maximization (EM) algorithm would also be complex [4]. A method for jointly performing the tasks of timing recovery and turbo equalization was proposed in [5], with complexity comparable to a conventional turbo equalizer. In this paper, we look at fundamental limits to the performance of timing recovery systems. We present a lower bound on the timing estimation error variance based on the Cramér-Rao bound. Due to the nature of the system model chosen, this bound is not achievable, and the closest we can get is by ML estimation, which is prohibitively complex. Therefore, to get a tighter practical bound, we look at the performance of a trained phase-locked loop (PLL) which gives a heuristic lower bound for the performance of iterative PLL-based receivers.