11 *4.6.2. Channel estimation and spatially correlation*

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7

8 9

6 filter at the GR the result given in [52] was used.

128 Contemporary Issues in Wireless Communications

12 infinite impulse response beamforming filter.

1 response beamforming filter for receiver discussed in [67]. The *SNR* was obtained by 2 averaging the respective *SNR*s over 1000 independent realizations of the typical urban 3 channel. For this purpose, in the case of FIR beamforming filter at the GR, the *SNR* given by 4 (159) was used and the corresponding beamforming filters at the GR were calculated using 5 the gradient algorithm discussed in Section 4.3. For infinite impulse response beamforming

10 **Figure 9.** Average *SNR* of MMSE linear equalization at the GR for beamforming with FIR and infinite 11 impulse response filters. The result for single antenna transmission is also indicated (the curve 4). IIR –

13 For comparison, we also show simulation results with FIR linear equalization filters at the GR of length *LF* 4*L* for FIR beamforming filters at the GR with 1 *Lg* 14 (the curve 3) 15 optimized for infinite impulse response linear equalization filters at the GR. These 16 simulation results confirm that sufficiently long FIR linear equalization filters at the GR 17 closely approach performance of infinite impulse response linear equalization filters at the 18 GR, which are necessary for (178) to be valid. As expected, the beamforming with infinite 19 impulse response beamforming filters at the GR constitutes a natural performance upper 20 bound for the beamforming with FIR beamforming filters at the GR. However, interestingly, for the typical urban channel the FIR beamforming filter of length 3 *Lg* 21 (the curve 2) is 12 We choose the 0.5 rate Low Density Parity Check (LDPC) code with a block length of 64800 13 bits, which is also adopted by DVB-S.2 standard [86]. QPSK modulation with Gray mapping is adopted in *NT NR* 4 MIMO system. Meanwhile, we set <sup>1</sup> <sup>2</sup> *<sup>σ</sup><sup>h</sup> NT NR* 14 , and *EP Eb* 15 . The channel is generated with coherence time of *NP N <sup>D</sup>* 85 MIMO vector 16 symbol intervals, and then a LDPC codeword is transmitted via 100 channel coherent time 17 intervals for QPSK modulation. For GR spatially correlated MIMO channel the GR array correlation matrix **Rr** 18 with the following elements is adopted [68]

$$\begin{cases} \mathbf{R\_{r}}(n,n) = 1, \mathbf{R\_{r}}(m,n) = \mathbf{R\_{r}^{\*}}(m,n), m, n = 1, 2, 3, 4; \\\ \mathbf{R\_{r}}(1,2) = \mathbf{R\_{r}}(2,3) = \mathbf{R\_{r}}(3,4) = 0.4290 + 0.7766 \, j; \\\ \mathbf{R\_{r}}(1,3) = \mathbf{R\_{r}}(2,4) = -0.3642 + 0.5490 \, j; \\\ \mathbf{R\_{r}}(1,4) = -0.4527 - 0.0015 \, j. \end{cases} \tag{195}$$

20 The performance comparison, in terms of *BER*, of the proposed soft-output MMSE GR, the 21 modified soft-output MMSE GR [83], the conventional soft-output MMSE GR [84], and the 22 conventional MMSE GR [85] is presented in Fig.10 for spatially independent MIMO channel 23 and GR receiver spatially correlated MIMO channel. Also, a comparison with the soft-24 output MMSE V-BLAST detector discussed in [73] is made. The proposed MMSE GR 25 outperforms all the existing schemes with considerable gain, especially for receiver 26 correlation MIMO channel scenario. The underlying reason of this improvement is that the 27 MMSE GR, by taking channel estimation error, decision error propagation and channel 28 correlation into account, can output more reliable LLR to channel decoder. As channel 29 estimation error is the dominant factor influencing the system performance under the lower 30 *SNR* region, it can observed that the *BER* of the conventional soft-output MMSE GR [84] is 31 slightly better than that of the modified soft-output MMSE GR in the case of spatially 32 independent MIMO channel.

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2 **Figure 10.** *BER* performance of different detectors under a) spatially independent MIMO channel and 3 b) receiver spatially correlated MIMO channel.
