## 5 Performance Evaluations and Discussion
### M-QAM 4-SSB Uncoded System Using Shadow Equalizer Evaluation and Complexity in Massive MIMO  
The MIMO system is key to modern wireless technology toward 5G. The main goal of MIMO application is to realize the diversity and spatial multiplexing. Diversity enables the system to achieve robust communications through transmission with an independent channel. In the other hand, the spatial multiplexing can save energy consumption by increasing the data rate and efficiently use bandwidth. Specifically, different from single antenna systems in multipath channels, the data rate and capacity of a channel in MIMO are significantly improved by using multiple receive and transmit antennas. Also, MIMO-based wireless communication systems can enable higher data-rate with less required transmission energy under the same throughput requirement compared to the single-input single-output system [25].  
In our proposal, the 4-SSB M-QAM performs slightly close to the performance of 16 QAM over the OFDM scheme. However, the 4-SSB M-QAM with shadow equalizer decreases the SNR compared to the previous scenario as we apply both the equalization and massive MIMO. This efficiency is demonstrated after 20 dB SNR. In the other hand, the 4-QAM OFDM has good performance because it carries only two bits (Figure 8).  
Figure 7: 4-QAM 4-SSB with turbo code system using shadow equalization compared to the relevant modulation schemes.  
Figure 9 shows the performance evaluations of our 4-SSB-based proposal and other relevant MIMO schemes in which the Rayleigh channel model is applied for the multipath fading channel environment.  
In this channel, the quadrature modulations of 4-QAM and 16-QAM over OFDM applied in 4-SSB are sensitive to the residual phase error, resulting in distances in quadrature-order modulation. The uniform distribution of data sources mapped by QAM Gray coding and the large number of Hilbert transform filter tabs in 4-SSB M-QAM can be approximated as a complex Gaussian distribution. Additionally, the different types of QAM Gray coding of symbols are bit mapped very close to one another. However, we still can obtain sufficient spectral efficiency.  
Next, we consider a more realistic MIMO scenario by showing the impact of the channel correlation. Particularly, we adopt a spatially correlated MIMO fading model using the Kronecker product model [26]. This model assumes the fading statistics of the transmit and receive antennas are independent. The correlation matrices are then computed based on the Bessel distribution correlation and the distance between antennas. In Figure 10, we take the distances between antennas as \(\mathrm{d}=0.3\lambda\). As the effect of the channel correlation is examined for the uncoded case, we show that the same results can be obtained in the case of non-correlated Rayleigh channel.  
Figure 8: 4-QAM 4-SSB shadow in massive MIMO compared to relevant schemes in Additive White Gaussian Noise (AWGN) environment.  
Figure 9: 4-QAM 4-SSB shadow in massive MIMO compared to the relevant modulation schemes in Rayleigh Channel environment.  
Finally, we apply 4-SSB using shadow equalizer in OFDM-GI as a new modulation to match the 5G requirements. Specifically, the discrete Fourier transform spread OFDM-GI (DFT-s-OFDM-GI) replaces cyclic prefix (CP) instead of the Guard Interval. In addition, the OFDM-GI can choose the time duration in a flexible way. Typically, this improves the performance of 4-SSB by using additional Hilbert transform for symbol generation. Hence, the Hilbert transform induces delay by 90 degrees, i.e., ISI is increased as well. Figure 11 shows an interesting result by investigating the evaluation history of 4-SSB.  
First, 4 bits in 4-QAM 4-SSB performs close to 2 bits of 4-QAM after 35 dB SNR. The remaining evaluations show that the proposal can reach Shannon boundary in the codec system since it performs well under the harsh condition of the uncoded environment.