Praise for Fundamentals of WiMAX

5.6 Shortcomings of Classical MIMO Theory 1815.5.3.2 Linear Precoding and PostcodingThe SVD illustratived how linear precoding and postcoding can diagonalize the MIMO channelmatrix to provide up to min( N dimensions to communicate data symbols through. Morer, Nt)generally, the precoder and the postcoder can be jointly designed based on such criteria as theinformation capacity [50], the error probability [21], the detection MSE [52], or the receivedSNR [51]. From Section 5.3.3, recall that the general precoding formulation isy = G(HFx + n), (5.64)where x and y are M× 1, the postcoder matrix G is M× N r , the channel matrix H is N , ther× Ntprecoder matrix F is N , and n is . For the SVD example, , G =U * t× M N r×1 M = min( Nr, Nt), and F = V.Regardless of the specific design criteria, the linear precoder and postcoder decompose theMIMO channel into a set of parallel subchannels as illustrated in Figure 5.15. Therefore, thereceived symbol for the ith subchannel can be expressed asy = ασ βx + βn , i =1, ⋯, M,i i i i i i i(5.65)where x iand y iare the transmitted and received symbols, respectively, with E | x i| = xasusual, are the singular values of H, and α i and β i are the precoder and the postcoder weights,respectively. Through the precoder weights, the precoder can maximize the total capacity by distributingmore transmission power to subchannels with larger gains and less to the others—referred to as waterfilling. The unequal power distribution based on the channel conditions is aprincipal reason for the capacity gain of linear precoding over the open-loop methods, such asBLAST. As in eigenbeamforming, the number of subchannels is bounded byσ i1 ( , ),2 ε≤ M ≤ min N (5.66)tNrwhere M =1 corresponds to the maximum diversity order, called diversity precoding inSection 5.3.3) and M = min( N achieves the maximum number of parallel spatial streams.t, Nr)Intermediate values of M can be chosen to provide an attractive trade-off between raw throughputand link reliability or to suppress interfering signals, as shown in the eigenbeamforming discussion.5.6 Shortcomings of Classical MIMO TheoryIn order to realistically consider the gains that might be achieved by MIMO in a WiMAX systems,we emphasize that most of the well-known results for spatial multiplexing are based on themodel in Equation (5.54) of the previous section, which makes the following critical assumptions.• Because the entries of H are scalar random values, the multipath is assumed negligible,that is, the fading is frequency flat.• Because the entries are i.i.d., the antennas are all uncorrelated.• Usually, interference is ignored, and the background noise is assumed to be small.