# Superimposed Perfect Binary Array-Aided Channel Estimation for OTFS Systems
## 4 Spectral Efficiency and Optimal Power Ratio
### Spectrum Efficiency Analysis  
The ultimate goal of designing superimposed PBA is to improve the SE. In OTFS systems, SE is given as [17]  
\[\text{SE}=(1-\eta)\text{log}_{2}(1+\text{SINR}_{\text{eff}}), \tag{40}\]  
where \(\eta\) denotes the pilot overhead and \(\text{SINR}_{\text{eff}}\) denotes the effective SINR for the OTFS system. In the proposed method, because the PBA used as the pilot is superimposed on the transmitted symbol matrix, it holds that \(\eta=0\) when compared to \(\eta=(2l_{m}+1)(4k_{m}+1)/MN\) for the integer Doppler condition and \(\eta=(2l_{m}+1)/M\) for the fractional Doppler condition in [13]. Thus, the superimposed PBA fundamentally improves the SE of an OTFS system. Next, we derive an expression for the effective SINR in SPBA-OTFS systems using the received signal decoupled in the DD domain. In the SPBA-OTFS system, the receiver decouples PBA and data by utilizing the estimated channel parameters in the DD domain. According to (9), the received symbol matrix \(\overline{\mathbf{Y}}\) after decoupling can be expressed as \[ \begin{split}\mathbf{\overline{Y}}&=\mathbf{Y}-\sum_{q=1}^{P} \tilde{I}_{q}e^{-j2\pi\frac{i\lambda_{k}q}{MN}}\mathbf{P}_{k_{q},l_{q}}\\ &=\sum_{q=1}^{P}h_{q}e^{-j2\pi\frac{i\lambda_{k}q}{MN}}\left( \mathbf{D}_{k_{q},l_{q}}+\mathbf{P}_{k_{q},l_{q}}\right)\\ &+\mathbf{W}-\sum_{q=1}^{P}\tilde{h}_{q}e^{-j2\pi\frac{i\lambda_ {k}q}{MN}}\mathbf{P}_{k_{q},l_{q}}.\end{split}  \tag{41}\]  
Recall from (36) that \(\tilde{I}_{q}=h_{q}+\bar{v}_{q}\), (41) can be rewritten as  
\[ \begin{split}\mathbf{\overline{Y}}&=\sum_{q=1}^{P} \left(\tilde{I}_{q}-\bar{v}_{q}\right)e^{-j2\pi\frac{i\lambda_{k}q}{MN}}\left( \mathbf{D}_{k_{q},l_{q}}+\mathbf{P}_{k_{q},l_{q}}\right)\\ &+\mathbf{W}-\sum_{q=1}^{P}\tilde{I}_{q}e^{-j2\pi\frac{i\lambda_ {k}q}{MN}}\mathbf{P}_{k_{q},l_{q}}\\ &=\sum_{q=1}^{P}\tilde{I}_{q}e^{-j2\pi\frac{i\lambda_{k}q}{MN}} \mathbf{D}_{k_{q},l_{q}}+\sum_{q=1}^{P}\left(-\bar{v}_{q}\right)e^{-j2\pi\frac {i\lambda_{k}q}{MN}}\mathbf{D}_{k_{q},l_{q}}\\ &+\sum_{q=1}^{P}\left(-\bar{v}_{q}\right)e^{-j2\pi\frac{i\lambda _{k}q}{MN}}\mathbf{P}_{k_{q},l_{q}}+\mathbf{W}.\end{split}  \tag{42}\]  
We use matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) to simplify (42), where \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) denote the effective signal, the interference introduced by data and the interference introduced by PBA, respectively. For the linear MMSE estimator, the estimation error is orthogonal to the observations [17]. Thus, the average effective SINR of each DD domain symbol is given by  
\[\text{SINR}_{\text{eff}}=\frac{\mathbb{E}\{\mathbf{A}\mathbf{A}^{*}\}}{\mathbb{ E}\{\mathbf{B}\mathbf{B}^{*}\}+\mathbb{E}\{\mathbf{C}\mathbf{C}^{*}\}+\mathbb{E}\{ \mathbf{W}\mathbf{W}^{*}\}}, \tag{43}\]  
where  
\[\mathbb{E}\{\mathbf{A}\mathbf{A}^{*}\}=MN\sum_{q=1}^{P}\sigma_{\tilde{I}_{q}} ^{2}(1-\beta)\rho, \tag{44}\]  
\[\mathbb{E}\{\mathbf{B}\mathbf{B}^{*}\}+\mathbb{E}\{\mathbf{W}\mathbf{W}^{*}\} =MN\left(\sum_{q=1}^{P}\sigma_{\tilde{v}_{q}}^{2}(1-\beta)\rho,+\sigma_{\omega} ^{2}\right). \tag{45}\]  
For \(\mathbb{E}\{\mathbf{C}\mathbf{C}^{*}\}\), we use the Cauchy-Schwarz inequality to derive its upper bound as follows:  
\[ \begin{split}\mathbb{E}\{\mathbf{C}\mathbf{C}^{*}\}& =\mathbb{E}\left\{\sum_{q=1}^{P}\bar{v}_{q}e^{-j2\pi\frac{i\lambda_{k}q}{MN}} \mathbf{P}_{k_{q},l_{q}}\sum_{q=1}^{P}\bar{v}_{q}^{*}e^{j2\pi\frac{i\lambda_{ k}q}{MN}}\mathbf{P}_{k_{q},l_{q}}^{*}\right\}\\ &\leq\mathbb{E}\left\{\sum_{q=1}^{P}\bar{v}_{q}\bar{v}_{q}^{*} \right\}\mathbb{E}\left\{\sum_{q=1}^{P}\mathbf{P}_{k_{q},l_{q}}\mathbf{P}_{k_ {q},l_{q}}^{*}\right\}\\ &=MNP\beta\rho\sum_{q=1}^{P}\sigma_{\tilde{v}_{q}}^{2}.\end{split}  \tag{46}\]By substituting (44), (45) and (46) into (43), the effective SINR is obtained as follows:  
\[