# Energy Harvesting for TDS-OFDM in NOMA-Based Underwater Communication Systems
## 5 Information Receiver for TDS-OFDM-NOMA
### UWA Channel Estimation  
The perfect channel estimation can provide a perfect removal of the IBI caused by the PN on the OFDM data blocks. After subtracting the IBI from the received OFDM data blocks, the TDS-OFDM will be equivalent to the BER performance of the conventional CP-OFDM and ZP-OFDM schemes. The differences in the received frame between the conventional TDS-OFDM, DPN-OFDM, and the CS-based OFDM used in this paper are shown in Figure 4.  
As shown in Figure 4, the received PN sequence of the TDS-OFDM schemes is affected by the IBI because of the OFDM data blocks; this mutual interference is due to the UWA channel multipath. This mutual interference affected the accuracy of the UWA channel estimation. The time-domain compressive sensing channel estimation technique for the TDS-OFDM by using the received PN sequence can be written as follows [7]:  
\[\mathbf{z}_{i}=\mathbf{\
abla}_{i}\mathbf{h}_{i}+\mathbf{v}_{i} \tag{32}\]  
where \(\mathbf{v}_{i}\) is the cumulative channel noise, and \(\mathbf{\
abla}_{i}\) is the sensing matrix used for compressive sensing. In this paper, the channel estimation is constricted based on the received PN sequence, and it can be written as:  
\[\mathbf{\
abla}_{i}=\begin{bmatrix}c_{i,0}&x_{i-1,N-1}&\cdots&x_{i-1,N-L+1}\\ c_{i,1}&c_{i,0}&\cdots&x_{i-1,N-L+2}\\ \vdots&\vdots&\ddots&\vdots\\ C_{i,L-1}&C_{i,L-2}&\cdots&C_{i,0}\\ C_{i,L}&C_{i,L-1}&\cdots&C_{i,1}\\ \vdots&\vdots&\ddots&\vdots\\ C_{i,M-1}&C_{i,M-2}&\cdots&C_{i,M-L}\\ \end{bmatrix}_{M\times L} \tag{33}\]  
In the compressive sensing theory, the propagation paths \(P\) are affected by the sparsity level indication of the communication channel. For accurate channel estimation, the sparsity channel indicator should be small. The sparsity channel indicator is affected not only by the number of channels taps \(P\); it is also affected by the channel length \(L\) and observation \(\mathcal{F}\), which is the number of rows. According to research work in [7], the sparsity indicator of the channel must satisfy the restricted isometry principle less (RIPless) condition for accurate channel estimation, and it can be represented in mathematical form as:  
\[P\leq\frac{\mathcal{F}}{4\mathfrak{C}_{0}\left(\log\left(\frac{L}{\mathcal{F} }\right)+1\right)} \tag{34}\]  
where \(\mathfrak{C}_{0}\) is a universal constant approximated by one, and it is normally adjusted according to the various situations of each compressive sensing matrix.  
In the TDS-OFDM (\(i-1\))th received frame, the compressive sensing matrix in Equation (33) is represented the previously transmitted block. The previously transmitted block in (33) interferes with the current TDS-OFDM symbols, while the previous TDS-OFDM data block does not contain the last samples of the received training sequence \(M-L+1\). In the compressive sensing-based IBI-free region, the last samples of the received PN training sequence are used for channel estimation, and the last \(F=M-L+1\) rows of the \(\boldsymbol{\Psi}_{i}\) used as in [7] for channel estimation as an observation matrix can be written as follows:  
\[\boldsymbol{\Phi}_{i}=\begin{bmatrix}c_{i,L-1}&c_{i,L-2}&c_{i,L-3}&\cdots&c_{i, 0}\\ c_{i,L}&c_{i,L-1}&c_{i,L-2}&\cdots&c_{i,1}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ c_{i,M-1}&c_{i,0}&c_{i,M-3}&\cdots&c_{i,\;M-L}\end{bmatrix}_{F\times L} \tag{35}\]  
The observation matrix size \(F\times L\) is determined by the received PN sequence. In the end, the compressive sensing equation of the free IBI region used in channel estimation will only be corrupted by the noise and can be written as follows:  
\[y_{i}=\boldsymbol{\Phi}_{i}\boldsymbol{h}_{i}+\boldsymbol{v}_{i} \tag{36}\]  
without any loss of generality, in this paper, we will apply the look-ahead backtracking orthogonal matching pursuit (LAB-OMP) algorithm proposed in [7] for channel estimation.