# Blind Turbo Equalization of Short CPM Bursts for UAV-Aided Internet of Things
## 4 EMLVA for Blind Channel Equalization
### The EMVA/EMLVA Blind Equalizer
It should be noted that under the condition of a short burst, the coded CPM system exhibits positive feedback in the process of outer iteration and convergence to a suboptimal solution by directly exchanging extrinsic information. Therefore, extrinsic information exchange methods play an important role in the convergence of the outer iteration, and the performance of equalization can be further improved by superior extrinsic information exchange methods. ### Complexity Analysis

The computational complexity of the EMV based on the BCA initializer mainly comes from the Viterbi-like algorithm of the E-step. Only considering this part, the complexity of the EMVA is \(O([2(D+1)^{2}SpM^{L+L_{c}-1}N])\), and the complexity after adding the outer iteration of turbo equalization is \(O([2(D+1)]^{2}SpM^{L+L_{c}-1}N+TSM^{L+L_{c}-1}N)\), where \(D\) is the channel memory length, \(L_{c}\) is the length terms of the symbol time, \(L\) is the memory length of the CPM frequency pulse, \(M\) is the base number of CPM, \(S\) is the maximum number of inner iterations, \(T\) is the maximum number of external iterations, and \(N\) is the sequence length. To reduce the calculation amount, the EMLVA can be implemented by applying the Lazy VA in the E-step. In the best case, the complexity can be reduced to \(O([2(D+1)]^{2}SN)\), and T-EMLVA has complexity of \(O([2(D+1)]^{2}SN+TSN)\). After adding the iteration stopping criterion, the number of outer iterations can be further reduced and the decoding delay can be effectively reduced. The average number of iterations of turbo equalization after adding the stopping criterion can be denoted as \(T_{ave}\), and the computational amount is \(O([2(D+1)]^{2}SN+T_{ave}SN)\). A detailed comparison of the complexity of the T-EMVA and the T-EMLVA is included in \[L(\mathbf{\theta}|\mathbf{\theta}^{(s)})=E_{a}(LLR|\mathbf{r},\mathbf{\theta}^{(s)})\approx \sum_{q=1}^{Q}\frac{P_{q}^{(s)}||\mathbf{r}-\mathbf{S}(\mathbf{\alpha})\mathbf{c}||^{2}}{N_{ 0}}-ln(N_{0})+d. \tag{19}\]1

## 5 Experimental Evaluation

### Experimental Setup

For the evaluation of the proposed blind turbo equalization, the parameters are summarized in Table 3 and the burst structure is shown in \[L(\mathbf{\theta}|\mathbf{\theta}^{(s)})=E_{a}(LLR|\mathbf{r},\mathbf{\theta}^{(s)})\approx \sum_{q=1}^{Q}\frac{P_{q}^{(s)}||\mathbf{r}-\mathbf{S}(\mathbf{\alpha})\mathbf{c}||^{2}}{N_{ 0}}-ln(N_{0})+d. \tag{19}\]2 GMSK with the modulation index h = 1/2 was taken as an example for the experiment and simulation. The channel was modeled as a two-ray Ricean channel model, which set the maximum delay of 7 us; to better simulate it in the hilly/mountainous scenarios for UAV-aided IoT systems, the Ricean factor was chosen as 29 dB. The sample time with a baud rate of 150 kHz was 3.3 us, and then, the symbol duration was \(T=6.6\) us. For the maximum delay 7 us, the channel length at the receiver was \(D+1=\lfloor 7/3.3\rfloor+1=3\) taps, while the channel length in terms of symbol duration was \(L_{c}=\lfloor D/\varepsilon\rfloor+1=2\); the sign \(\lfloor\rfloor\) represents rounding down. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Standard Equalization** & **Turbo Equalization** \\ \hline EMVA & \(O([2(D+1)]^{2}SpM^{L+L_{c}-1}N)\) & \(O([2(D+1)]^{2}SM^{L+L_{c}-1}N+TSpM^{L+L_{c}-1}N)\) \\ EMLVA & \(O([2(D+1)]^{2}SN)\) & \(O([2(D+1)]^{2}SN+TSN)\) \\ EMLVA with HDA & \(O([2(D+1)]^{2}SN)\) & \(O([2(D+1)]^{2}SN+T_{ave}SN)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Complexity comparison of the algorithms. ### Simulation Results

#### 5.2.1 System Parameter Optimization

A serially concatenated CPM (SCCPM) setup with iterative decoding was considered in the numerical simulations. In view of the insufficiency of interleaving for the short burst, the optimized parameters of the convolutional code (CC) and interleaver need to be selected. The number of states and the free distance of a CC can affect the interleaving gain and the performance of the iterative detection.