# Blind Turbo Equalization of Short CPM Bursts for UAV-Aided Internet of Things
## 4 EMLVA for Blind Channel Equalization
### The EMVA/EMLVA Blind Equalizer
As seen in Figure 6b, the BCA-initializer-based EMLVA can ensure good convergence of the blind equalization with a fixed inner iteration of three. More outer iterations did not improve the performance significantly after about five iterations, which can be viewed as convergence. On the contrary, when a single initializer was used, the BER curves decreased slowly as \(E_{b}/N_{0}\), and there was not a considerable performance improvement in the iterative process. The performance loss can be explained by the poor convergence to track changing channels in short bursts. Figure 7 analyses the normalized-mean-squared error (NMSE) of the EMVA and EMLVA based on the BCA initializer, as well as the NMSE of the EMLVA using a single initializer and the NMSE based on the training bits method. It can be seen that the proposed EMLVA equalizer with the BCA initializer significantly outperformed the EMVA in complexity while keeping a similar estimation accuracy as the EMVA. Moreover, the NMSE of the EMLVA based on a single initializer had poor convergence, and there was no decreasing trend at high \(E_{b}/N_{0}\). At the same time, the NMSE curves of the data-aided method with few training bits exhibited poor convergence and estimation accuracy, although there was a decreasing trend. #### 5.2.3 Improved Exchange Methods of Extrinsic Information and Stopping Criterion

Direct exchange is an original method, which can be further improved by weighted exchange. In weighted exchange, the extrinsic information \(L_{E}\) is sent to the weighted function module before being delivered to the other SISO module. The weighted function is expressed as:

\[W(L_{E})=\alpha L_{E}*exp(-\beta|L_{E}|). \tag{28}\]

In order to evaluate the effect of different weighting coefficients, the simulation was implemented with values of \(\alpha\in[0.8,1.0]\) and \(\beta\in[0.001,0.01]\). The BER performance with various weighted coefficient combinations of \(\alpha\) and \(\beta\) is shown in Figure 8a-c. Figure 8d shows the BER as a function of \(E_{b}/N_{0}\) for different extrinsic information, including weighted extrinsic information, average extrinsic information, and direct methods, with or without the HDA stopping criterion. The results show that there was little loss of performance of the algorithm with HAD criterion and the combination of \(\alpha=0.9\) and \(\beta=0.01\) had the best BER performance. Figure 7: NMSE as a function of \(E_{b}/N_{0}\) for training-based channel estimation. EMVA with single fixed initializer; EMLVA and EMVA with BCA initializer. #### 5.2.4 The Effect of Channel Length Overestimation on the Performance

Finally, the effect of channel length overestimation on the performance was investigated. Up till this point, the channel length was assumed known. However, in practice, this is not the case. The BER performance of the EMLVA with different initializer schemes is analyzed in Figure 9a when the assumed channel length is correct (\(\hat{N}_{c}=0\)) and when it is overestimated (\(\hat{N}_{c}=1,2,3\)). The results illustrate the apparent robustness of the EMLVA with the BCA initializer compared to the EMLVA with a single initializer. Figure 9b shows the number of expansions performed by the EMLVA and EMVA as a function of \(E_{b}/N_{0}\), where the channel overestimation length is expressed in symbol time. It is easy to see that the number of states traversed by the EMVA increased exponentially with the overestimation length, while the number of states searched by the EMLVA remained constant for high \(E_{b}/N_{0}\). Even in the event of severe channel length overestimation, the complexity of the EMLVA can be decoupled from the memory length above a certain \(E_{b}/N_{0}\). Therefore, Figure 9b shows the superiority of the EMLVA proposed in this paper in terms of complexity, which can meet the computational requirements of low complexity for IoT nodes. Figure 8: The performance of different exchange methods of extrinsic information with or without a stopping criterion.