# Blind Turbo Equalization of Short CPM Bursts for UAV-Aided Internet of Things
## 4 EMLVA for Blind Channel Equalization
### The EMVA/EMLVA Blind Equalizer
When the set of initializers has been traversed, a set of estimated parameters is obtained, and then, the optimal estimated parameters need to be selected. For the \(k\)th initializer, the EMV equalizer converges to \(\hat{\mathbf{\theta}}_{\mathbf{k}}=(\hat{\mathbf{c}},\hat{\mathbf{N}}_{0})_{k}\), and the evaluation of the likelihood function can be expressed as

\[L(\mathbf{\theta}_{\mathbf{k}})=-Ln\hat{\mathbf{N}}_{0,k}-\frac{\hat{\mathbf{\eta}}_{k}}{\hat {\mathbf{N}}_{0,k}}+ln\vec{L}_{k}. \tag{23}\]

Figure 4: In (**a**), EMLVA blind equalizer block diagram. In (**b**), turbo EMLVA blind equalizer block diagram. where

\[\widetilde{L}_{k}=\sum_{q=1}^{Q}exp[\frac{\hat{\eta}_{k}-\hat{d}_{k,q}}{N_{0,k}}], \tag{24}\]

\[\hat{d}_{k,q}=||r-S(\mathbf{a}_{q})\hat{\mathbf{e}}_{k}||^{2}, \tag{25}\]

\[\hat{\eta}_{k}=mind_{k,q}. \tag{26}\]

The best estimated parameters can be selected by maximizing \(L(\theta_{k})\), i.e.,

\[L(\mathbf{\theta}|\mathbf{\theta}^{(s)})=E_{a}(LLR|\mathbf{r},\mathbf{\theta}^{(s)})\approx \sum_{q=1}^{Q}\frac{P_{q}^{(s)}||\mathbf{r}-\mathbf{S}(\mathbf{\alpha})\mathbf{c}||^{2}}{N_{ 0}}-ln(N_{0})+d. \tag{19}\]0

### The Turbo EMLVA Blind Equalizer and Positive Feedback

Figure 4b shows that the detector can further improve performance through turbo equalization in Section 3.1, which can effectively improve the bit error rate performance and convergence of the algorithm. A turbo equalization EMV algorithm based on the BCA initializer is presented in Algorithm 3. To distinguish the iteration of turbo equalization from the iteration of the EM algorithm, the turbo equalization between CPM-SISO and Decoder-SISO is denoted as the outer iteration, and the iteration of the EM algorithm is called the inner iteration. ```
1:Input: the received signal \(\mathbf{r}\)
2: Set the maximum outer iteration equalization times \(T\)
3:for\(t=0\) to \(T\)do
4: With the current initialization to \(\mathbf{\theta}^{(0)}=(\epsilon^{(0)},N_{0}^{(0)})\), run Algorithm 2
5: Compute an estimate to the extrinsic LLRs \(L_{E}(b_{t}|(\mathbf{r}))\). Feed \(L_{E}(b_{t}|(\mathbf{r}))\) to the channel decoder
6: From the channel decoder per-bit soft-output, recompute LLRs for each symbol \(L_{D}(\alpha_{k})\)
7:endfor
8:Output: Output the soft decision output detection results
```

**Algorithm 3** The EMLVA blind turbo equalizer (T-EMLVA). (1) A set of initializers \(\mathbf{\theta}_{k}^{(0)}=(\epsilon^{(0)},N_{0}^{(0)})_{k},k=1,2,...,K\), and select the \(k\)th initial value \(\mathbf{\theta}^{(0)}=(\epsilon^{(0)},N_{0}^{(0)})_{k}\). (2) Set the maximum iteration number of turbo equalization \(T_{max}\), with the initial outer iteration \(T=0\); set the maximum iteration number of iterations of the EM algorithm \(S_{max}\) with the initial inner iteration \(s=0\). (3) With the current initializer \(\mathbf{\theta}\), the received signal \(r\) and the a priori information \(L_{D}(u)\) returned by the convolutional code (initially 0), run Algorithm 2 to obtain the Q survivor paths \(\left\{\mathbf{a}_{q}\right\}_{q=1}^{Q}\), the path metric \(\left\{\Lambda_{q}\right\}_{q=1}^{Q}\), and the soft information \(\left\{L_{E}^{q}(\mathbf{\alpha})\right\}_{q=1}^{Q}\) of each path. (4) After traversing all the initializers, the optimal estimated parameter is selected by the convergence criterion in Equation (27) for the blind equalization, and the corresponding soft symbol information \(L_{E}(\mathbf{\alpha})\) is demapped and deinterleaved to obtain the soft information \(L_{E}(\mathbf{b}_{t})\) of the information bits, which is input into the channel decoder as a priori information. (5) When \(T<T_{max}\), an extrinsic LLR on the coded bits \(L_{D}(\mathbf{b})\) is mapped and interleaved again and delivered back to the CPM blind equalization as the updated a priori probability. Steps 3 to 5 are repeated for a given maximum number of iterations \(T_{max}\). For turbo equalization, the inner iteration using different Viterbi algorithms is denoted as T-EMVA and T-EMLVA, respectively.