## 3 Proposed Channel Estimation Schemes
### Scheme I: Channel Estimation Using DBT-MC-FTN Pilots  
In this method, a DBT-MC-FTN pilot is applied to each subframe. As shown in part (I) marked in Figure 1a, the pilot symbol is mapped to the subcarriers as the modulated data symbols do in the time domain, i.e., the pilot and data symbols are time division multiplexing. Figure 3 depicts the time-frequency grid of the DBT-MC-FTN system where the pilot symbol is located at the first symbol in one subframe. The packing factors in the DBT-MC-FTN pilots are the same as the data symbols, therefore the spectral efficiency will not be affected. Another advantage that we can employ is that, once the FPF and TPF are known, the self-introduced ICI and ISI are deterministic and can be described mathematically by **C**.  
Next, the single tap frequency-domain channel estimation (FD-CE) scheme and time-domain channel estimation (TD-CE) scheme have been applied, respectively.  
In the case of single tap FD-CE, the estimated CIR can be obtained via LS algorithm as given below:  
\[\mathbf{\hat{h}}_{F}=\mathbf{X}^{-1}\mathbf{R}. \tag{12}\]  
The mean squared error (MSE) can be calculated by  
\[\begin{split} MSE_{FD-CE}&=E\{(\mathbf{h}-\mathbf{X }^{-1}\mathbf{R})^{H}(\mathbf{h}-\mathbf{X}^{-1}\mathbf{R})\}=E\{(\mathbf{X}^ {-1}\mathbf{Z}_{\mathbf{\Phi}^{*}})^{H}(\mathbf{X}^{-1}\mathbf{Z}_{\mathbf{ \Phi}^{*}})\}\\ &=E\{\mathbf{Z}_{\mathbf{\Phi}^{*}}^{H}(\mathbf{X}\mathbf{X}^{H} )^{-1}\mathbf{Z}_{\mathbf{\Phi}^{*}}\}.\end{split} \tag{13}\]  
Figure 3: DBT-MC-FTN time-frequency resource block of one subframe: (**a**) DBT-MC-FTN transmission subframe structure; (**b**) the time-frequency grid of a DBT-MC-FTN symbol.  
According to Equation (7), the estimated CIR in the time domain with the LS algorithm can be calculated as  
\[\mathbf{\hat{h}}_{T}=(\mathbf{\Theta}^{H}\mathbf{\Theta})^{-1}\mathbf{\Theta}^{H} \mathbf{y}, \tag{14}\]  
where \(\mathbf{H}_{c}\) can be constructed by taking \(\mathbf{\hat{h}}_{T}\) as the first column and cyclic shifting \(\mathbf{\hat{h}}_{T}\). Then, inverse \(\mathbf{H}_{c}^{-1}\) of \(\mathbf{H}_{c}\) can be utilized to equalize the received data symbol to compensate for the channel effects as follows:  
\[\mathbf{y}_{eq}=\mathbf{H}_{c}^{-1}\mathbf{y}=\mathbf{H}_{c}^{-1}\mathbf{H}_{ c}\mathbf{\Phi}\mathbf{x}+\mathbf{H}_{c}^{-1}\mathbf{z}. \tag{15}\]  
Similar to Equation (13), the MSE of TD-CE is given by  
\[MSE_{TD-CE} =E\{(\mathbf{h}-\mathbf{\hat{h}}_{T})^{H}(\mathbf{h}-\mathbf{ \hat{h}}_{T})\} \tag{16}\] \[=E\{(\mathbf{h}-(\mathbf{\Theta}^{H}\mathbf{\Theta})^{-1}\mathbf{ \Theta}^{H}\mathbf{y})^{H}(\mathbf{h}-(\mathbf{\Theta}^{H}\mathbf{\Theta})^{ -1}\mathbf{\Theta}^{H}\mathbf{y})\}\] \[=E\{[(\mathbf{\Theta}^{H}\mathbf{\Theta})^{-1}\mathbf{\Theta}^{H }\mathbf{z}]^{H}[(\mathbf{\Theta}^{H}\mathbf{\Theta})^{1}\mathbf{\Theta}^{H} \mathbf{z}]\}\] \[=E\{\mathbf{z}^{H}\mathbf{\Theta}[\mathbf{\Theta}^{H}\mathbf{ \Theta}(\mathbf{\Theta}^{H}\mathbf{\Theta})]^{-1}\mathbf{\Theta}^{H}\mathbf{z }]\}.\]  
As shown in Equation (7), \(\mathbf{H}_{c}\) is a Toeplitz matrix whose first column is \(\mathbf{h}=[h_{1},h_{2},...,h_{Q}]^{T}\). As a result, \(\mathbf{H}_{eq}\) is a diagonal matrix where there is no packing in time domain or frequency domain, i.e., orthogonal system. Thus, a simple single-tap FD-CE can be implemented by Equation (12). In this case, we have \(\mathbf{\Theta\Theta}^{H}=N_{s}\mathbf{I}_{N_{s}}\) and it can further be obtained that \(MSE_{FD-CE}=\frac{\sigma_{F}^{2}}{\sigma_{K}^{2}}\) and \(MSE_{TD-CE}=\frac{\sigma_{F}^{2}}{N_{s}}\). However, when it turns to the DBT-MC-FTN system with two-dimensional packing, there are off-diagonal elements in \(\mathbf{H}_{eq}\) due to the ISI/ICI introduced by time/frequency domain packing and ICI introduced by the filter trailing. Since the pilot symbols are modulated and packed as the DBT-MC-FTN symbol in a non-orthogonal manner, in such case, the single-tap FD-CE algorithm is not applicable anymore. Thus, the channel estimation will be processed in the time domain.