## 3 Virtual Space-Time Diversity Turbo Equalization
### Bidirectional Combination  
The virtual time-reversal mirror (VTRM) is designed to reduce the error propagation (EP) of equalizers with feedback input. The bidirectional combination scheme based on VTRM consists of two individual FS-SEs with opposite direction. The input of the backward FS-SE is time-reversed of received signal, as \(\{\overset{\leftrightarrow}{z}_{n}\}\leftarrow\{z_{-n}\}\), while the input of the forward FS-SE is \(\{z_{n}\}\). Assuming that the noise of input of the bidirectional FS-SE is independent Gaussian noise for the memoryless character of the Gaussian process, the noise of the outputs can be approximatively regarded as independent Gaussian noise. The SNR of soft symbols processed by linear bidirectional combination is improved, so that the bidirectional combination scheme can be considered as a kind of time diversity technology.  
The linear combined soft symbol of the equalizer output is given as  
\[\overset{\leftrightarrow}{d}_{n}=\eta_{1}\overset{\rightarrow}{d}_{n}+\eta _{2}\overset{\leftarrow}{d}_{n} \tag{22}\]  
where \(\mathcal{CN}(\overset{\rightarrow}{d}_{n};x_{n},\overset{\rightarrow}{ \sigma}_{d}^{2})\) and \(\mathcal{CN}(\overset{\leftarrow}{d}_{n};x_{n},\overset{\leftarrow}{\sigma}_ {d}^{2})\) are the distributions of bidirectional FS-SEs, \(\eta_{1}\in\mathbb{R}^{+}\) and \(\eta_{2}\in\mathbb{R}^{+}\) are the combination factors with \(\eta_{1}+\eta_{2}=1\). Assuming that the estimate variances of bidirectional FS-SEs are the same as \(\overset{\rightarrow}{\sigma}_{d}^{2}=\overset{\leftarrow}{\sigma}_{d}^{2}= \sigma_{v}^{2}\), the output SNR of the forward FS-SE is defined as:  
\[\overrightarrow{\text{SNR}}=\frac{\sigma_{x}^{2}}{\sigma_{v}^{2}} \tag{23}\]  
where \(\sigma_{x}^{2}\) is the variance of \(\mathbf{x}\). Corresponding to Equation (22), the SNR of \(\overset{\leftrightarrow}{d}_{n}\) is written as [20]:  
\[\overrightarrow{\text{SNR}}=\frac{\mathbb{E}|\eta_{1}x_{n}+\eta_{2}x_{n}|^{2}} {\mathbb{E}|\eta_{1}d_{n}+\eta_{2}d_{n}|^{2}-\mathbb{E}|\eta_{1}x_{n}+\eta_{ 2}x_{n}|^{2}}\approx\frac{(\eta_{1}+\eta_{2})^{2}\sigma_{x}^{2}}{(\eta_{1}^{2 }+\eta_{2}^{2}+2\xi\eta_{1}\eta_{2})\sigma_{v}^{2}} \tag{24}\]  
where \(\xi\in[0,1]\) is the cross-correlation coefficient. According to the assumption that the noise of bidirectional FS-SEs is approximatively independent, i.e., \(\xi\to 0\), we can obtain \(\overrightarrow{\text{SNR}}\approx\frac{1}{\eta_{1}^{2}+(1-\eta_{1})^{2}} \overrightarrow{\text{SNR}}\geqslant\overrightarrow{\text{SNR}}\). This conclusion demonstrates the performance improvement of the bidirectional combination scheme.  
In fact, the noise of the bidirectional FS-SEs is not strictly independent. To acquire the property of \(\xi\), we use the real oceanic data set of the Pengcheng Laboratory for analysis. The data set adopts binary phase shift keying (BPSK) modulation, a 10 kHz carrier frequency, 1 kSymbol/s and 40 ms LFM frame header with 8 k\(\sim\)12 kHz bandwidth. The cross-correlation coefficient is calculated as [23]:  
\[\xi=\frac{\overset{\leftrightarrow}{\sigma}_{d}^{2}-\eta_{1}^{2}\overset{ \rightarrow}{\sigma}_{d}^{2}-\eta_{2}^{2}\overset{\leftarrow}{\sigma}_{d}^{2 }}{2\eta_{1}\eta_{2}\overset{\leftarrow}{\sigma}_{d}^{2}\overset{ \rightarrow}{\sigma}_{d}} \tag{25}\]  
where \(\overset{\leftrightarrow}{\sigma}_{d}^{2}=\frac{1}{N}\sum_{n=1}^{N}| \overset{\leftrightarrow}{d}_{n}-x_{n}|^{2}\), \(\overset{\leftarrow}{\sigma}_{d}^{2}=\frac{1}{N}\sum_{n=1}^{N}|\overset{ \leftarrow}{d}_{n}-x_{n}|^{2}\), \(\overset{\rightarrow}{\sigma}_{d}^{2}=\frac{1}{N}\sum_{n=1}^{N}|\overset{ \rightarrow}{d}_{n}-x_{n}|^{2}\).  
We compared three kinds of situations with different fractionally spaced orders, i.e., 1/2, 1/4, 1/8. The lengths of FFF and SIC are both 20 and \(\eta_{1}=\eta_{2}=0.5\). Figure 5 shows that cross-correlation coefficient decreases with the fractionally spaced order. The average cross-correlation coefficients are separately 0.397, 0.311, 0.133 for the 1/2, 1/4, 1/8 fractionally spaced orders.