## 3 Virtual Space-Time Diversity Turbo Equalization
### FS-SE with Anti-Doppler Module  
The time-domain channel equalizer structure, as shown in Figure 2, is a loop-locked scheme to overcome the fast time-varying Doppler effect. The loop-locked anti-Doppler time-domain channel equalizer consists of an anti-Doppler module (ADM) and a fractionally spaced soft interference cancellation (SIC)-based equalizer (FS-SE) module. The ADM utilizes an adjustable scheme to solve the dynamic residual Doppler effect of the individual frame. The FS-SE module is exploited to eliminate ISI. Figure 1: Structure of the proposed VSTD-TEQ for UWA communication systems. A two-step method is exploited to accomplish Doppler spread compensation. The original dominant Doppler spread of the received signal is first compensated by a Farrow structure interpolator. Assuming that the main Doppler factor estimated by synchronizing the signal through the matched filtering techniques is \(\hat{\mathbf{\lambda}}\) and the total Doppler factor estimate is \(\mu\), the main time-domain spread caused by the Doppler effect of the received signal is compensated with an interpolator. Without losing generality, we adopt Lagrange interpolation (\(N\)-segment \(M\)-order polynomial) with Farrow structure [40] in this work. After down conversion, the frontforward input of FS-SE is given as follows  
\[z_{n}=\sum_{i=0}^{M}\sum_{j=0}^{N}\frac{\hat{g}_{m_{n}-j}c_{i,j}}{(1+\mu)^{i}} \text{exp}\{-j2\pi f_{c}(kT_{c}/K_{s}+\hat{\tau})\} \tag{9}\]  
where \(c_{i,j}\) is the coefficient of the interpolator, \(m_{n}\) is the floor integer index of resampling time instant, \(\hat{\tau}\) is the time delay of frame synchronization. The FS-SE contains \(K_{s}\)-order parallel feedforward filters (FFFs) and a SIC filter for compensation of the linear characteristics of the UWA channel. According to the order of fractional space [34], \(z_{n}\) is downsampled and allocated to \(K_{s}\) FFFs. The output of FS-SE is given as  
\[\hat{d}_{n}=\sum_{k=1}^{K_{s}}\mathbf{\xi}_{k,n}^{H}\mathbf{z}_{k,n}e^{-j\hat{\phi}_{n }}-\mathbf{\mathbf{b}}_{n}^{H}\tilde{\mathbf{\mathbf{x}}}_{n} \tag{10}\]  
where the length of FFF \(\mathbf{\mathbf{f}}_{k,n}\) is \(N_{f,1}+N_{f,2}+1\), the length of SIC is \(N_{b,1}+N_{b,2}+1\). \(\mathbf{\mathbf{z}}_{k,n}=[z_{k,n+N_{f,1}},\cdots,z_{k,n-N_{f,2}}]^{T}\) and \(\tilde{\mathbf{\mathbf{x}}}_{n}=[\mathbf{\mathbf{x}}_{n-N_{b,1}},\cdots,\mathbf{\mathbf{x }}_{n-1},0,\tilde{\mathbf{\mathbf{x}}}_{n+1},\cdots,\mathbf{\mathbf{x}}_{n+N_{b,2}}]^{T}\) are the inputs of \(k\)-th FFF and SIC, respectively. \(\hat{\phi}_{k}\) is the phase compensation of DPLL. Especially, the SIC will degenerate into the normal feedback filter of the decision-feedback equalizer when \(N_{b,2}=0\). The FS-SE output using combined taps is rewritten as  
\[\hat{d}_{n}=\mathbf{\mathbf{w}}_{n}^{H}\mathbf{\mathbf{u}}_{n} \tag{11}\]  
where \(\mathbf{\mathbf{w}}_{n}=[\mathbf{\mathbf{f}}_{1,n}^{T}e^{-j\hat{\phi}_{n}},\cdots,\bm {\mathbf{f}}_{K_{s},n}^{T}e^{-j\hat{\phi}_{n}},-\mathbf{\mathbf{b}}_{n}^{T}]^{T}\),\(\mathbf{\mathbf{u}}_{n}=[\mathbf{\mathbf{z}}_{1,n}^{T},\cdots,\mathbf{\mathbf{z}}_{K_{s},n}^{T}, \mathbf{\mathbf{x}}_{n}^{T}]^{T}\), and their lengths are both \(N_{eq}=K_{s}(N_{f,1}+N_{f,2}+1)+N_{b,1}+N_{b,2}+1\). The adaptive update of \(\mathbf{\mathbf{w}}_{n}\) will be discussed in the following section. The residual Doppler effect contributes to phase rotation of the FS-SE input signal. We can obtain the instantaneous phase error of the equivalent phase detector (PD) [41; 42]:  
\[\theta_{n}=\mathfrak{I}\{p_{n}(x_{n}+q_{n})\}\approx\mathfrak{I}\Big{\{}d_{n} x_{n}^{*}\Big{\}} \tag{12}\]  
where \(p_{n}=\sum_{k=1}^{K_{s}}\mathbf{\mathbf{f}}_{k,n}^{H}\mathbf{\mathbf{z}}_{k,n}e^{-j \hat{\phi}_{n}}\), \(q_{n}=\mathbf{\mathbf{b}}_{n}^{H}\tilde{\mathbf{\mathbf{x}}}_{n}\). Assuming \(\mathbb{E}[p_{n}q_{n}^{*}]=0\), we can obtain Equation (12) by avoiding the calculation of \(p_{n}\) and \(q_{n}\). The expression of Equation (12) is similar to the phase estimate result in [43].