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The transmitted signal (in baseband) [12,13] is given by N −1 xa (t) = sn h a (t − kTs ), n=0 where • • • {s0 , . . , s N −1 } are the transmitted symbols belonging to ASK, PSK or QAM constellations or any linearly precoded scheme, with N being the sequence length used for estimating the synchronization (sync) parameters; Ts is the symbol period which is assumed to be known throughout this chapter. For more information about the blind estimation of Ts , the reader is referred to [17]; h a (t) is the shaping filter that classically is a square-root Nyquist filter.

But not the synchronization step since, except in very infrequent cases, the synchronization step duration is always much smaller than the channel coherence time. Before going further, we recap the optimal symbol detector when synchronization is perfect (τ0 = 0, φ0 = 0, and f 0 = 0) [12,13,18,19]. If the information symbols are equally likely, the maximum likelihood detector is the optimal one in the sense of error probability minimization. ,N −1 ). ,N −1 R dt/2N0 . 2 N −1 sˆn = arg 2 sn h(t − nTs ) dt.

2 Synchronization in Flat Fading Channel 25 step (feasible thanks to the training sequence) in order to ensure low data detection error. In order to avoid the calculation of the derivative at each symbol period (notice that in the blockwise approach the number of iterations and thus of derivative calculations is much smaller than the number of samples), we estimate it with z τˆ (m−1) + (n) − z τˆ (m−1) − (n) ∂z τ (n) N = N , ∂τ τˆ (m−1) 2 N where is a design parameter that must be carefully adjusted.