Praise for Fundamentals of WiMAX

5.4 Beamforming 171and a unique solution for the weight vector is readily obtained as[ ]w = 0.3034 + j0.1966 0.3932 0.3034 − j0.1966 .T(5.47)Figure 5.9 shows the beam pattern using this weight vector. As expected, the beamformer hasunity gain for the desired user and two nulls at the directions of two interferers. Since the beamformercan place nulls in the directions of interferers, the DOA-based beamformer in this exampleis often called the null-steering beamformer [27]. The null-steering beamformer can be designedto completely cancel out interfering signals only if the number of such signals is strictly less thanthe number of antenna elements. That is, if N r is the number of receive antennas, N r−1 independentinterferers can be canceled. 5 The disadvantage of this approach is that a null is placed in thedirection of the interferers, so the antenna gain is not maximized at the direction of the desireduser. Typically, there exists a trade-off between interference nulled and desired gain lost. A moredetailed description on the DOA-beamformer with refined criterion can be found in [27, 38].Thus far, we have assumed that the array response vectors of different users with correspondingAOAs are known. In practice, each resolvable multipath is likely to comprise severalunresolved components coming from significantly different angles. In this case, it is not possibleto associate a discrete AOA with a signal impinging the antenna array. Therefore, the DOAbasedbeamformer is viable only in LOS environments or in environments with limited localscattering around the transmitter.5.4.2 EigenbeamformingUnlike DOA-based beamforming, eigenbeamforming does not have a similar physical interpretation.Instead of using the array-response vectors from AOAs of all different users, eigenbeamformingexploits the channel-impulse response of each antenna element to find array weightsthat satisfy a desired criterion, such as SNR maximization or MSE (mean squared error) minimization.By using channel knowledge at the transmitter, eigenbeamforming exploits the eigendecomposedchannel response for focusing the transmit signal to the desired user even if there arecochannel interfering signals with numerous AOAs. Because eigenbeamforming is a mathematicaltechnique rather than a physical technique for increasing the desired power and suppressingthe interference signals, it is more viable in realistic wireless broadband environments, whichare expected to have significant local scattering. When we refer to an eigenchannel in this section,we are referring to the complex channel corresponding to an eigenvalue in the channelmatrix, which can be accessed by precoding with the (right) eigenvector of the channel matrix.Consider a MIMO eigenbeamforming system using N tantennas for transmission and N rantennas for reception in a flat-fading channel. It is assumed that there are L effective cochannel5. In some special cases, it may be possible to cancel more than N r – 1 interferers, such as the specialcase in which a third interferer was at an angle of 2π/3 or 7π/6 as in Figure 5.9.