# Effective Receiver Design for MIMO Visible Light Communication with Quadrichromatic LEDs
## 4 Numerical Results  
In this section, we give the results for the proposed and conventional receivers in the VLC system using QLEDs that are presented in a room measuring \(4\times 4\times 3\) m. We consider a \(16\times 16\) MIMO system as there are four RAGB LEDs and four multi-color receivers. The positions of QLEDs are fixed at \((1,1,3)\) m, \((1,-1,3)\) m, \((-1,1,3)\) m, \((-1,-1,3)\) m, respectively. Meanwhile, the receivers are located at two different sets of locations in two scenarios. In Location 1, the receivers are placed near the center as the coordinates are \((0.2,0.3,0)\) m, \((0.2,0.1,0)\) m, \((0,0.3,0)\) m, \((0,0.1,0)\) m. In Location 2, the receivers are positioned far from the center with the locations are \((1,1.1,1)\) m, \((1,0.9,1)\) m, \((0.8,1.1,1)\) m, \((0.8,0.9,1)\) m.  
Moreover, each QLED employs the 4-CSK constellation [7]. Consequently, there are 256 possible symbols for the system of four QLEDs. For simulation with transmission rate of \(R=4,6\) bps, \(M=16,64\) symbols are randomly chosen from the set of 256 possible symbols. Moreover, in this paper, we set \(\Phi_{1/2}=60\deg,\theta_{\text{fov}}=60\deg,A_{r}=1\text{ cm}^{2},\delta=1\text{ A/W}\). The signal-to-noise ratio (SNR) is defined as \(SNR=\frac{1}{2\sigma^{2}}\) as transmit power is assumed to be normalized to unity [3]. To find the set of optimal normal vectors for the receivers, the MATLAB optimization toolbox CVX is employed [11]. A set of 200 initial normal vectors \(\mathbf{U}\) is utilized to find the optimal orientation for the receivers. On the other hand, as mentioned in previous studies [3; 5; 6; 7; 8; 9; 10], the conventional receiver generally employs \([0\quad 0\quad 1]^{T}\) as the fixed normal vector since the receiver orientation is usually assumed to be orthogonal to the plane that contains the transmitters.  
Figure 3 shows the performance comparison between the proposed, conventional, and the adapted HR [10] for \(R=4\) bps in Location 1 with different crosstalk values. Moreover, the proposed receiver achieves different SNR gains for various \(\epsilon\) values. When \(\epsilon=0\), the proposed receiver can achieve 3 dB SNR gain while the HR and conventional receiver have almost the same performances. In comparison with the conventional receiver, the proposed receiver can improve the performance by 4 dB in case of \(\epsilon=0.01\) while the gap is only 1 dB in case of \(\epsilon=0.1\). Similarly, in comparison with the HR, the proposed receiver can achieve large SNR gains in the low and medium SNR regimes. The gaps are narrow in the high SNR regimes, but the proposed receiver still maintains better performance in comparison with the HR. Furthermore, the lower the crosstalk level, the better performances of all receivers. Meanwhile, the HR is a low complexity receiver that does not require the orientation adjustment when changing location but can not ensure a favorable and optimal performance in every location.  
In Figure 4, we compare the proposed and conventional receivers in both locations when \(R=6\) bps and \(\epsilon=0.01,0.1\). Since Location 1 is near the center of the room, the correlation between the channel coefficients is high. Therefore, the performance of the conventional receiver in Location 1 is bad. The proposed receiver can significantly alleviate this problem by alternating the normal vectors with the SNR gains that are nearly 4 dB and 2 dB. In addition, in Location 2, the conventional receiver with straight upward normal vectors can not receive sufficient signal power from all QLEDs as in Location 1 and hence the performance becomes worst. However, the proposed receiver with an adequate tuning of the normal vectors can enhance the signal received from all QLEDs. This combines with the inherently low correlation of the MIMO channel in Location 2, dramatically improving the performance. In particular, the receivers in Location 2 have the best performance, and the SNR gains are around 9 dB for both cases of \(\epsilon\).