# Blind Turbo Equalization of Short CPM Bursts for UAV-Aided Internet of Things
## 2 Related Work  
In practical communications, the signals are transmitted over the fading channel and the channel response is unknown. In [14], a generalized pilot symbol-aided demodulation method based on the idea of inserting data-dependent symbols periodically was proposed for CPM in a flat Rayleigh fading channel. An optimal front-end filter was developed based on the mean-squared error (MSE) in the channel estimation process. Then, the channel estimates generated by the interpolation filter, together with the received signal, are input into a coherent CPM demodulator using the Viterbi algorithm. In frequency-flat fast-fading channels, Ref. [15] provided a data-aided channel estimation algorithm with local B-splines, and the results showed that there exists a minimum sampling interval proportional to the normalized fading rate for pilot insertion. However, when short bursts are considered, the data-aided channel estimation method can significantly increase the overhead-to-payload ratio. Similarly, low-complexity frequency-domain equalization for CPM [27; 28; 29] requires the addition of a cyclic prefix or unique words, which can also increase the overhead-to-payload ratio for short bursts. As an alternative, blind channel equalization can recover the signal directly, without a training sequence. The author in [16] applied the Tong-Xu-Kailath algorithm to CPM by extracting the second-order statistics of the signal for channel estimation. The eigenvector method was used to identify the channel from a fourth-order cross-cumulant matrix under the GSM channel in [17], combined with turbo estimation. However, when applied to a low number of symbols, the statistical moments did not provide accurate channel estimation. In [18; 30], the author developed a nonlinear signal model for GMSK rather than the conventional finite impulse response model. The information symbols were obtained by Bayesian inference based on Markov chain Monte Carlo (MCMC) with implicit channel estimation. CPM and the multipath channel can construct a joint trellis, which can be represented by a finite state machine (FSM). Therefore, a forward adaptive SISO (FA-SISO) [19], which considers the channel correlation in only one direction, was proposed for MSK, which \begin{table}
\begin{tabular}{c c l} \hline \hline
**Classification** & **Ref.** & **Contribution/Methodology** \\ \hline MCMC & [18] & A nonlinear signal model for GMSK and information symbols with implicit channel estimation by MCMC are developed. \\ \hline \multirow{6}{*}{Adaptive equalization} & [19] & A forward adaptive SISO that considers the channel correlation in only one direction is proposed for MSK. \\ \cline{2-3}  & [20] & A variety of the reduced state FA SISO is proposed. \\ \cline{2-3}  & [21] & The thresholds of the RS-A-SISO algorithms are obtained by the density evolution technique. \\ \cline{2-3}  & [22] & Derivation of the forward/backward adaptive algorithm. \\ \cline{2-3}  & [23] & Derivation of the generalized forward/backward adaptive algorithm. \\ \hline \multirow{6}{*}{HMM} & [24] & The BBW algorithm, as well as two variants, are proposed for CPM. \\ \cline{2-3}  & [25] & A stochastic ML blind channel estimation is developed, and an approximate Cramer-Rao bound for CPM is derived. \\ \cline{2-3}  & [26] & The Viterbi algorithm is applied within the EM algorithm. \\ \hline \multirow{6}{*}{FDE} & [27] & The single-carrier frequency-domain equalization is used in the CPM signal for the first time. \\ \cline{2-3}  & [28] & Laurent decomposition is used to realize traditional equalization (linear and decision feedback) and turbo equalization in the frequency domain. \\ \cline{1-1} \cline{2-3}  & [29] & More iterative gain without matrix inversion. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._replaces the unknown channel by the least-mean-squared error for each hypothesis branch symbol. Then, due to the high complexity of FA-SISO, the author in [20] proposed various reduced-state A-SISO (RS-A-SISO) algorithms for complexity reduction at the same time.