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

Channel Estimation 157The mean square error is given byJ n,i = E{|ε n,i | 2 }. (4.48)The optimal filter in the sense of minimizing J n,i with the minimum mean square error criterionis the two-dimensional Wiener filter. The filter coefficients of the two-dimensionalWiener filter are obtained by applying the orthogonality principle in linear mean squareestimation,E{ε n,i H˘n ∗ ′′ ,i ′′} =0, ∀{n′′ ,i ′′ }∈ n,i . (4.49)The orthogonality principle states that the mean square error J n,i is minimum if the filtercoefficients ω n ′ ,i ′ ,n,i, ∀{n ′ ,i ′ }∈ n,i are selected such that the error ε n,i is orthogonalto all initial estimates H˘n ∗ ′′ ,i ′′ , ∀{n ′′ ,i ′′ }∈ n,i . The orthogonality principle leads to theWiener–Hopf equation, which states thatE{H n,i H˘n ∗ ′′ ,i ′′} =∑ω n ′ ,i ′ ,n,iE{ ˘{n ′ ,i ′ }∈ n,iH n ′ ,i ′H˘n ∗ ′′ ,i ′′}, ∀{n′′ ,i ′′ }∈ n,i . (4.50)With Equation (4.44) and by assuming that N n ′′ ,i ′′ has zero mean and is statisticallyindependent from the pilot symbols S n ′′ ,i ′′, the cross-correlation function E{H ˘n,iHn ∗ ′′ ,i ′′ }is equal to the discrete time–frequency correlation function E{H n,i H˘n ∗ ′′ ,i ′′ },i.e.thecrosscorrelationfunction is given byθ n−n ′′ ,i−i ′′ = E{H n,iHn ∗ ′′ ,i′′}. (4.51)The auto-correlation function in Equation (4.50) is given byφ n ′ −n ′′ ,i ′ −i ′′ = E{ ˘H n ′ ,i ′˘H ∗ n ′′ ,i′′}. (4.52)When assuming that the mean energy of all symbols S n,i including pilot symbols is equal,the auto-correlation function can be written in the formφ n ′ −n ′′ ,i ′ −i ′′ = θ n ′ −n ′′ ,i ′ −i ′′ + σ 2 δ n ′ −n ′′ ,i ′ −i ′′. (4.53)The cross-correlation function depends on the distances between the actual channelestimation position n, i and all pilot positions n ′′ ,i ′′ , whereas the auto-correlation functiondepends only on the distances between the pilot positions and, hence, is independent ofthe actual channel estimation position n, i. Both relations are illustrated in Figure 4-20.Inserting Equations (4.51) and (4.52) into Equation (4.53) yields, in vector notation,θ T n,i = ωT n,i, (4.54)where is the N tap × N tap auto-correlation matrix and θ n,i is the cross-correlation vectorof length N tap . The vector ω n,i of length N tap represents the filter coefficients ω n ′ ,i ′ ,n,i