Multi-Carrier and Spread Spectrum Systems: From OFDM and MC ...

146 Implementation IssuesThese sequences r(k) could be known in the receiver by transmitting, for instance, twoconsecutive reference symbols (M = N c ) as proposed by Moose [60], or one can exploitthe presence of the guard time (M = L g ) [78].The maximization of the above LLF can be done in two steps:– A first maximization can be performed to find the frequency error estimate ˆ f error ,– In a second step, the value of the given frequency error estimate is exploited forfinal maximization to find the timing error estimate ˆτ error .The maximization of f error is given by the partial derivation ∂LLF(f error , τ error )/∂ f error =0, which results infˆerror =− 12π ∠γ(τ error) + z =− 12πm+M−1 ∑k=mm+M−1∑k=mIm [r(k)r ∗ (k + N c )]Re [r(k)r ∗ (k + N c )]+ z, (4.32)where z is an integer value.By inserting ˆ f error in Equation (4.30), we obtainLLF ( ˆ f error ,τ error ) =|γ(τ error )|−ρ(τ error ), (4.33)and maximizing Equation (4.33) gives us a joint estimate of ˆ f error and ˆτ error :fˆerror =− 12π ∠γ(ˆτ error),ˆτ error = arg(maxτ error{|γ(τ error )|−ρ(τ error )}. (4.34)Note that in case of M = N c (i.e. two reference symbols), | fˆerror | < 0.5,z= 0, and notiming error τ error = 0(m = 0), one obtains the same results as Moose [60] (see Figure 4-12).The main drawback of the Moose maximum likelihood frequency detection is the smallrange of acquisition, which is only half of the sub-carrier spacing F s .Whenf error →0.5F s , the estimate fˆerror may, due to noise and the discontinuity of arctangent, jump to−0.5. When this happens, the estimate is no longer unbiased and in practice it becomesuseless. Thus, for frequency errors exceeding one-half of the sub-carrier spacing, aninitial acquisition strategy, coarse frequency acquisition, should be applied. To enlargethe acquisition range of a maximum likelihood estimator, a modified version of thisestimator was proposed in Reference [12]. The basic idea is to modify the shape of theLLF.The joint estimation of frequency and timing error using the guard time may be sensitivein environments with several long echoes. In the following section, we will examinesome approaches for time and frequency synchronization which are used in several implementations.