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

158 Implementation Issuesˆ H n,i . Hence, the filter coefficients of the optimum two-required to obtain the estimatedimensional Wiener filter areω T n,i = θ T n,i −1 , (4.55)when assuming that the auto- and cross-correlation functions are perfectly known andN tap = N grid . Since in practice the auto-correlation function and cross-correlation functionθ n,i are not perfectly known in the receiver, estimates or assumptions about thesecorrelation functions are necessary in the receiver.4.3.1.2 Two Cascaded One-Dimensional FiltersTwo-dimensional filters tend to have a large computational complexity. The choice of twocascaded one-dimensional filters working sequentially can give a good tradeoff betweenperformance and complexity. The principle of two cascaded one-dimensional filtering isdepicted in Figure 4-21. Filtering in the frequency direction on OFDM symbols containingpilot symbols, followed by filtering in the time direction on all sub-carriers is shown. Thisordering is chosen to enable filtering in the frequency direction directly after receivinga pilot symbol bearing OFDM symbol and, thus, to reduce the overall filtering delay.However, the opposite ordering would achieve the same performance due to the linearityof the filters.The mean square error of the two cascaded one-dimensional filters working sequentiallyis obtained in two steps. Values and functions related to the first filtering are marked withthe index [1] and values and functions related to the second filtering are marked with theindex [2] . The estimates delivered by the first one-dimensional filter areH ˆ [1]n,i ′ =∑ω [1] ˘ n ′ ,nH n ′ ,i′. (4.56){n ′ ,i ′ }∈ n,i ′The filter coefficients ω [1]n ′ ,nonly depend on the frequency index n. This operation isperformed in all ⌈N s /N t ⌉ pilot symbol bearing OFDM symbols. The estimates deliveredfreq.time1st filtering on pilot symbolbearing OFDM symbols02nd filtering oneach sub-carrierdata symbolpilot symbol0N c − 1N s − 1Figure 4-21Two cascaded one-dimensional filter approach