## 3 Synchronization and Channel Estimation for TIBWB-OFDM
### Channel Estimation  
After the frame synchronization procedure, the CSI is acquired. Three algorithms were developed for estimating the CSI:  
* ZC preamble-based channel estimation (Algorithm A);
* data-based channel estimation (Algorithm B); and
* combined ZC and data channel estimation (Algorithm C).  
```
1:Input: \(\mathbf{X}\), \(\mathbf{S_{ZC}}\) and \(\delta_{decision}\).
2:Output: \(y_{sync}\);
3:\(y_{auto\_corr}\leftarrow\) Perform autocorrelation of \(\mathbf{S_{ZC}}\);
4:\(y_{corr}\leftarrow\) Perform cross-correlation between \(\mathbf{S_{ZC}}\) and \(\mathbf{y}\);
5:\(threshold\leftarrow\delta_{decision}\times\mathbf{max}\{|y_{auto\_corr}|\}\);
6:\(y_{thold}\gets y_{corr}\geq threshold\);
7:\(y_{index}\leftarrow\) Find index peaks in \(y_{thold}\)
8:for\(iteration=1:2L_{p}\)do
9:\(y_{sync}\leftarrow\) Tracking peaks in \(y_{index}\) within an \(\mathcal{L}\) sliding window and stores the higher peak value;
10:endfor
```  
**Algorithm A**  
Knowing the estimated \(y_{sync}\), received samples corresponding to \([ZP,ZC]\) are used to estimate the channel. An \(FFT\) of length \(N_{ZP}+N_{p}\) is applied to the received samples and divided by the corresponding transmitted samples, i.e., a LS estimator is used according to  
\[\widetilde{H}_{k(zc),(N_{ZP}+N_{p})}=\frac{FFT_{N_{ZP}+N_{p}}\{\widetilde{s}_{ zc}\}}{FFT_{N_{ZP}+N_{p}}\{\widetilde{s}_{zc}\}}, \tag{22}\]  
where \(\widetilde{s}_{zc}\) represents the received preamble sequence. The \(N_{ZP}\) sequence length is always larger than the CIR, thus enabling the reception of \(\widetilde{s}_{ZC}\) without ISI. The estimated low-resolution (in frequency) CSI is then upconverted by interpolation to the length of the TIBWB-OFDM block.  
\[\widetilde{H}_{k(zc),(N_{b})}=FFT_{(N_{b})}\Big{\{}IFFT_{N_{ZP}+N_{p}}\Big{\{} \widetilde{H}_{k(zc),(N_{ZP}+N_{p})}\Big{\}}\Big{\}}, \tag{23}\]  
where \(\widetilde{H}_{k(zc),(N_{b})}\) represents the estimated channel constituted by \(N_{b}\) samples.  
Algorithm B  
If the IB-DFE is adopted, estimated data \(\widehat{X}_{k}^{(l-1)}\) at iteration \(l-1\) can be used to obtain an estimation of the channel to be used at iteration \(l\). This is given by  
\[\widetilde{H}_{k(data)}=\frac{Y_{k}}{\widehat{X}_{k}^{(l-1)}}, \tag{24}\]  
where \(\widehat{X}_{k}^{(l-1)}\) denotes the frequency-domain signal estimated at the previous \((l-1)\) iteration with \(N_{b}\) samples.  
A refined strategy for achieving successively better estimations can be used by leveraging on estimations from Algorithm B. Once CSI is obtained with Algorithm A on the basis of the known preamble sequence, it makes no sense to disregard such information and consider only Algorithm B on the basis of estimated data. Instead, Algorithm C considers a weighted mean between the ZC channel estimated sequence (Algorithm A), and the data sequence estimated sequence (Algorithm B) can be applied as follows  
\[\widetilde{H}_{k(data+zc)}=\frac{\left(\frac{\widetilde{H}_{k(data)}}{\sigma_ {(data)}^{2}}+\frac{\widetilde{H}_{k(zc)}}{\sigma_{(zc)}^{2}}\right)}{\left( \frac{1}{\sigma_{(data)}^{2}}+\frac{1}{\sigma_{(zc)}^{2}}\right)}, \tag{25}\]  
where \(\sigma_{(zc)}^{2}\) and \(\sigma_{(data)}^{2}\) individually represent the mean square error between the estimated ZC and data sequence, respectively, and the original channel. This relation is given by  
\[\sigma_{(zc/data)}^{2}=\mathbf{E}\bigg{\{}\left|H_{k}-\widetilde{H}_{k(zc/data )}\right|^{2}\bigg{\}}. \tag{26}\]