## 6 Simulations and Results
### Simulations  
We perform Monte Carlo simulations in a typical communication system, and assume that the signal generating parameter is uniformly distributed as well as the channel gain. Some of these parameters are listed in Table 1.  
We use the BER, convergence rate, and mutual information to show the accuracy of our CEVTI. In our pilot scheme, all transmitters transmit at the same time frequency and experience the same Rayleigh-fading channel. Furthermore, we assume there is no correlation between the transmitting antennas and the receiving antennas.  
We compare CEVTI with five channel estimation algorithms: LS, LLMSE, joint distributed CE, CE for FDD, LDAMP and VMP. All the variational-based algorithms use the same initialization, i.e., a pilot-based channel estimation and a MIMO channel model.  
Figure 1 shows the impact of noise variance on the estimation of the channel coefficient. It is intuitive that the BER increases with the noise variance at an acceleration rate. Please note that the sensor noise variance also has a huge impact on the estimation accuracy, and is more difficult to correct using CE algorithms, unless very large data samples are used. However, CEVTI displays higher accuracy compared with others, which means that CEVTI guarantees better local optimum.  
Figure 2 shows the BER of the signal-to-noise ratio (SNR, i.e., \(E_{b}/N_{0}\)). We iterate 10 times to simplify the comparison. The results show that CEVTI has a better transmitter performance than the five other existing CE algorithms. The BER of CEVTI is significantly lower than LLMSE-based algorithms. In addition, the convergence rate of CEVTI is non-degenerate compared to other iterative CE algorithms. We can see that LDAMP also works well due to its computational power and use of large training data. The CEVTI algorithm is also expected to produce a local minimum BER, as it can make the best use of local data samples, even when the sample size is small.  
Figure 3 shows the convergence behavior of CEVTI compared with LS, LLMSE, joint distributed CE, CE for FDD, LDAMP and VMP under different SNR. The results show that CEVTI converges faster and smoother than other CE algorithms. The reason is, as explained before, VT has guaranteed convergence and smooth descent. We can see from Figure 3, joint distributed CE and VMP also behave well due to the application of VI. For instance, at \(SNR=15\) dB, CEVTI converges in around 15 iterations, and joint distributed CE and VMP converge in 17 and 16 iterations, respectively. The simulated executing time of CEVTI is also better, showing that CEVTI has satisfying computational complexity.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Meaning** & **Value** \\ \hline \(L_{p}\) & pilot number & 32 \\ \(N\) & number of users & 8 \\ \(M\) & number of antennas & 4 \\ \(r_{ij}\) & receiver /’s signal toward user \(i\) & \\ \([\alpha_{1},\beta_{1}]\) & signal prior & [0.5,1] \\ \([\alpha_{2},\beta_{2}]\) & user prior & [0.4,0.6] \\ \(e\) & convergence tolerance & \(10^{-6}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameter notations and values.  
Figure 1: Noise Variance Impact.  
Figure 3: Average number of iterations until convergence.  
Figure 2: BER comparison.  
Figure 4 shows the mutual information comparison with LS, LLMSE, joint distributed CE, CE for FDD, LDAMP and VMP under different SNR. The mutual information is between users and BS antennas, which is a good indicator of the impact of channel estimation accuracy on the throughput of the average bandwidth. The results show that CEVTI has non-degenerate throughput compared with others. For instance, at \(SNR=20\) dB, CEVTI's mutual information is 41, same as VMP and joint CE, slightly higher than others, showing that CEVTI has theoretical satisfying network throughput.