composition in (3.37) is guaranteed to exist. P. Singh and E. Sharma of the unitary matrix Qm is obtained as a solution of the following optimization problem: s.t. QmQ = IN,- (3.43) For an orthogonal pilot matrix Bm (BmBH = 2PMIN ) [49], the solution of the optimization problem above is expressed as [50] (3.44) The matrices and VQ above are obtained using the SVD as follows: (3.45) Upon employing the estimate of whitening and unitary matrices, the semi-blind estimate of CSI matrix Hm is obtained as (3.46) H E C N, x N N, be the MIMO channel matrix, which is obtained as H=[Ho,H1,...,Hn-1] (3.47) The semi-blind estimate of HH can be calculated as HSBSSs : , (3.48) The algorithmic form of the semi-blind scheme is given in Algorithm 1. 3.4.3 MSE Gain of the Semi-Blind Estimate over the LS Estimate From (3.35), the pilot-based LS estimate of the CSI matrix Hm is given as One can obtain the LS estimate of the MIMO CSI matrix Hm as [24] HLS,m = (3.49) 3 FBMC: A Waveform Candidate for Beyond 5G Algorithm 1: Algorithmic form of the semi-blind estimator Input: Virtual pilot matrix Bm, Observation matrix Ym, Observation vectors , for Noise variance on and power Pd Output: Estimate Hm 1 Obtain the estimate , 2 Compute 3 Obtain the estimate Wm 4 Compute (WHYMBH) 5 Obtain the estimate 6 Obtain the estimate Hm WOOD 7 return: The estimate Hm Here the operation gives the pseudo-inverse of Bm [51]. As shown in [52], the MSE of the LS CSI estimator is (3.50) It is worth mentioning that the CRLB of the LS estimator equals the channel estimation error covariance, because HLS,m is the minimum variance unbiased estimate. The LS estimate of the MIMO CSI matrix in (3.47) is (3.51) The MSE in HLS is calculated as The CRLB for the channel estimation per parameter using the LS CSI estimator is determined as NN,Nr which can be seen to remain unchanged with N and Nr. Coming to the semi-blind estimator, the unitary matrix Qm constrained as QHQm = IN, - = Thus, one can utilize the constrained CRLB framework in [53] for benchmarking the MSE of t