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

156 Implementation Issueswhere S n,i and N n,i are the transmitted symbols and the noise components respectively.The pilot symbols are written as S n ′ ,i′, where the frequency and time indices at locationsof pilot symbols are marked as n ′ and i ′ . Thus, for equally spaced pilot symbols weobtainn ′ = pN f , p = 0,..., ⌈ ⌉N c /N f − 1 (4.41)andi ′ = qN t , q = 0,...,⌈N s /N t ⌉ − 1, (4.42)assuming that the first pilot symbol in the rectangular grid is located at the first sub-carrierof the first OFDM symbol in an OFDM frame. The number of pilot symbols in an OFDMframe results in⌈ ⌉⌈ ⌉Nc NsN grid =. (4.43)N f N tPilot symbol aided channel estimation operates in two steps. In the first step, the initialestimate H˘n ′ ,i′ of the channel transfer function at positions where pilot symbols are locatedis obtained by dividing the received pilot symbol R n ′ ,i′ by the originally transmitted pilotsymbol S n ′ ,i ′,i.e.H˘n ′ ,i ′ = R n ′ ,i ′= H nS ′ ,i ′ + N n ′ ,i ′. (4.44)n ′ ,i ′ S n ′ ,i ′In the second step, the final estimates of the complete channel transfer function belongingto the desired OFDM frame are obtained from the initial estimates H˘n ′ ,i ′ by twodimensionalinterpolation or filtering. The two-dimensional filtering is given byHˆn ′ ,i ′ =∑ω n ′ ,i ′ ˘,n,i{n ′ ,i ′ }∈ n,iH n ′ ,i′, (4.45)where ω n ′ ,i ′ ,n,i is the shift-variant two-dimensional impulse response of the filter. Thesubset n,i is the set of initial estimates H˘n ′ ,i ′ that is actually used for estimation of Hˆn,i .The number of filter coefficients isN tap =‖ n,i ‖ N grid . (4.46)In the OFDM frame illustrated in Figure 4-20, N grid is equal to 12 and N tap is equal to 4.Two-Dimensional Wiener FilterThe criterion for the evaluation of the channel estimator is the mean square value of theestimation errorε n,i = H n,i − Hˆn,i . (4.47)