# Blind Turbo Equalization of Short CPM Bursts for UAV-Aided Internet of Things
## 3 System Model and Problem Description  
### Communications System Model  
The communications system with blind turbo equalization is represented in Figure 1. It can be divided into three parts: transmitter, channel model, and blind turbo equalization:  
(1) Transmitter: A sequence of message bits \(\mathbf{m}=(m_{0},m_{1},m_{2},...,m_{K})\) is encoded with the channel encoder into codeword \(\mathbf{b}=(b_{0},b_{1},b_{2},...,b_{V})\) and interleaved as \(\mathbf{u}=(u_{0},u_{1},u_{2},...,u_{V})\). Then, the block codewords are input to the M-ary CPM modulation module, which consists of the continuous-phase encoder (CPE) and memoryless modulator (MM). The baseband representation of CPM is written as  
\[s(t,\mathbf{\alpha}_{n})=\sqrt{\frac{E_{s}}{T}}\exp(j\mathbf{\phi}(t,\mathbf{\alpha}_{n}) ),t\in[0,nT]. \tag{1}\]  
where \(E_{s}\) and \(T\) represent the symbol energy and symbol duration, respectively. \(\mathbf{\alpha}_{n}\) is the CPM symbol sequence, containing \(n+1\) symbols. The information-bearing phase is defined as  
\[\mathbf{\phi}(t,\mathbf{\alpha}_{n})=2\pi h\sum_{i=0}^{n}\alpha_{i}q(t-iT). \tag{2}\]\(h\) is the modulation index, defined as the quotient of two relatively prime integers. \(q(t)\) is the phase pulse obtained by the integral of the phase pulse \(g(t)\). \(L\) is the length of the frequency pulse, where \(L=1\) is the full-response CPM and \(L>1\) is the partial response. (2) Channel: The CPM transmitted over the channel is affected by the fading channel and Gaussian white noise; the continuous representation of the received signal is  
\[r(t)=s(t,\mathbf{\alpha})c(t)+w(t). \tag{3}\]  
In digital processing, it is necessary to use a discrete-time representation. The sampling time is noted as \(T_{s}=T/\varepsilon\), where \(\varepsilon\) is the number of sampling numbers per symbol. Therefore, the discrete-time representation of CPM, the received signal, and the noise can be defined as:  
\[\mathbf{s}_{n}=[s_{0},s_{1},s_{2}...,s_{nt-1}]^{T}, \tag{4}\]  
\[\mathbf{r}_{n}=[r_{0},r_{1},r_{2}...r_{nt-1}]^{T}, \tag{5}\]  
\[\mathbf{w}_{n}=[w_{0},w_{1},w_{2}...,w_{nt-1}]^{T}. \tag{6}\]  
The CPM signal for the \(i\)th path is defined as  
\[\mathbf{s}_{n-i}=[\mathbf{0}_{i},s_{0},w_{1},w_{2}...,s_{nt-i-1}]^{T}. \tag{7}\]  
where \(\mathbf{0}_{i}\) is a row vector containing \(i\) zero elements and discrete channel length \(N_{c}+1\). We define  
\[\mathbf{S}_{n}(\mathbf{\alpha})=[\mathbf{s}_{n},\mathbf{s}_{n-1},...,\mathbf{s}_{n-N_{c}}]^{T}. \tag{8}\]  
where \(\mathbb{C}^{n\times l}\) represents an \(n\)byl complex matrix. The channel response is \(\mathbf{c}=[c_{0},c_{1},...,c_{N_{c}}]^{T}\), and then, the discrete-time representation of the received signal in the form of a matrix can be written as  
\[\mathbf{r}_{n}=\mathbf{S}_{n}(\mathbf{\alpha}_{n})\mathbf{c}+\mathbf{w}_{n}. \tag{9}\]  
(3) Channel estimation and equalization: The CPM and multipath channel can build a joint trellis, which can be regarded as an FSM. According to Bayesian theory, the posterior probability is  
\[s(t,\mathbf{\alpha}_{n})=\sqrt{\frac{E_{s}}{T}}\exp(j\mathbf{\phi}(t,\mathbf{\alpha}_{n}) ),t\in[0,nT]. \tag{1}\]0  
Since the receiver does not have a priori information about the channel, the probability density function \(p(\mathbf{c})\) can be regarded as a constant. The received signal is known at the receiver, and as a result, \(p(\mathbf{r})\) is also a constant. Therefore, we can write \(p(\mathbf{\alpha},\mathbf{c}|\mathbf{r})\sim p(\mathbf{r}|\mathbf{\alpha},\mathbf{c})p(\mathbf{\alpha})\), where  
\[s(t,\mathbf{\alpha}_{n})=\sqrt{\frac{E_{s}}{T}}\exp(j\mathbf{\phi}(t,\mathbf{\alpha}_{n}) ),t\in[0,nT]. \tag{1}\]1  
With the perfect channel state information (CSI), maximum likelihood (ML) detection requires maximizing Equation (11). However, in practice, the channel response is unknown to the receiver and channel estimation is required. The FSM can be described by a hidden Markov model because the input symbols are independent of each other.