put multiple-output (MIMO) system. The linear receiver with matrix W is computed to get an estimate du = WHy However, this approach may lead to an over-complicated matrix, which makes the hardware realization infeasible. In order to maintain low complexity, the receiver matrix needs to have the structure W, u = where 1(hea) is a diagonal matrix for channel equalization, and is a GFDM matrix. We denote this receiver as GFDM-based receiver. Accordingly, the OFDM channel equalizer function can be reused with slight modification in computing the equalizer coefficients. Moreover, small-size FD-GFDM demodulation needs to be performed after the equalization. 4.3.2.1 LMMSE GFDM-Based Receiver For notation simplicity, we drop the index and focus on the model of the received FD-GFDM signal: y = A d Ad + v. Assuming uncorrelated noise, R = , = Rd = E[dd"] is the correlation matrix of the data symbols, and À is non-singular. The linear minimum mean squared error (LMMSE) receiver can be written as (4.27) 4 Generalized Frequency Division Multiplexing: Unified Multicarrier Framework Therefore, the LMMSE receiver can be decoupled into channel equalizer Heq and zero-forcing (ZF) demodulation, where From (4.21), it is clear that B is a GFDM matrix, and therefore, the demodulator win- dow W1x[k,m] = [Wtx[k,m]]-1. However, Heq is not necessarily diagonal depending on Rx = ÄR. The diagonal equalization corresponds to the element-wise equalization of the signal y[n] = h|n|x[n]+i[n],where E[X[n]²] Rx[n,n]. Accordingly, In the special case when Rd ESIN and À is orthogonal, Rx = ESIN, which produces the exact LMMSE, because (4.28) This is the case where the extended GFDM modulation is used as orthogonal precoding. On the other hand, when conventional GFDM is used to reduce the OOB, either the correlation matrix Rd is not a scaled identity or the modulation matrix is non-orthogonal, the diagonal equalization with ZF demodulation is an approximation of LMMSE. The computation of Rx[n,n] is performed offline for different configurations and 