# Energy Harvesting for TDS-OFDM in NOMA-Based Underwater Communication Systems
## 2 TDS-OFDM System Model  
In this section, the basic properties of TDS-OFDM are reviewed for our discussion. TDS-OFDM is the key technology in the Chinese standard of digital terrestrial multimedia/broadcasting (DTMB) and digital television terrestrial broadcasting (DTTB) [19]. The TDS-OFDM was proposed to avoid sending the pilot signal used in both CP-OFDM, ZP-OFDM, and NGI-OFDM for energy and bandwidth efficiency. However, TDS-OFDM energy and spectrum efficiency can cause significant bit error rate (BER) performance deterioration due to IBIs between the training sequence and OFDM data blocks [20]. The dual pseudorandom noise OFDM (DPN-OFDM) [21] and zero-pseudorandom noise OFDM (ZPN-OFDM) [22] were proposed to simplify the equalization and avoid the IBIs between OFDM data blocks and the training sequence. Although high BER improvement is achieved using DPN-OFDM and ZPN-OFDM, they waste channel bandwidth by using guard intervals twice in DPN-OFDM and ZPN-OFDM, making these schemes unfit for the limited UWA channel bandwidth [7]. In [7], the authors discussed how to avoid the IBI by estimating a semi-perfect UWA CSI using the IBI-free region and mitigating UWA channel interference by using compressive sensing techniques. The detailed comparison of frame structures among CP-OFDM, ZP-OFDM, and TDS-OFDM signals in time domains is shown in Figure 1, where the CP used by CP-OFDM is replaced by zeros in ZP-OFDM and by a known PN training sequence in TDS-OFDM. The frequency-domain symbol vector \(\mathbf{s}_{N}=\left[s_{0},s_{1},\cdots,s_{N-1}\right]^{T}\) is transmitted in the TDS-OFDM multicarrier system over the \(N\) subcarrier. The \(i\)-th time-domain transmission vector of the TDS-OFDM is:  
\[\mathbf{x}_{i}=\left\{x_{i,k}\right\}_{k=0}^{N-1}=\mathbf{F}_{N_{i}}^{H}\mathbf{s}_{N_{i}} \tag{1}\]  
where \(\mathbf{F}_{N}\) is the \(N\) point fast Fourier transform (FFT) matrix with the \((m,n)\)-th element \(\mathbf{F}_{N}(m,n)=\frac{1}{\sqrt{N}}e^{-j2\pi mn/N}\). \(k\) is the \(k\)-th element of the data block. The guard interval with length \(M\) is represented by \(c_{M}=[c_{0},c_{1},\cdots,c_{M-1}]^{T}\) in the conventional TDS-OFDM scheme. The total transmitted TDS-OFDM frame is defined as:  
\[\mathbf{s}_{i}=\left[\mathbf{s}_{i,0},\mathbf{s}_{i,1},\cdots,\mathbf{s}_{i,M+N-1}\right]^{T}= \left[\mathbf{x}_{i},\mathbf{c}_{i}\right]_{M+N-1} \tag{2}\]  
Let the transmitted PN sequence of the TDS-OFDM be \(\left\{c_{i,q}\right\}_{q=0}^{M-1}\). The TDS-OFDM transmitted symbol is denoted as \(a(n)\) and can be expressed as:  
\[a(n)=\left\{ \begin{array}{c}\frac{1}{N}\sum_{k=0}^{N-1}S(k)e^{j2\pi kn/N}\; \;n\in[0,N-1]\\ \left\{c_{i,k}\right\}_{k=0}^{M-1}\;n\in[N,N+M-1]\end{array} \right. \tag{3}\]  
This TDS-OFDM frame is transmitted over the UWA multi-path channel. The channel can be modeled by a quasi-static \(L_{cl}\)-th order finite impulse response (FIR) filter with channel impulse response (CIR) \(\left\{h_{i,k}\right\}_{k=0}^{L_{cl}-1}\). Assume the length of \(M\) is larger than the maximum channel tap delay, i.e., \(M\geq L_{cl}\). The \(i\)-th received OFDM frame \(\left\{r_{i,k}\right\}_{k=0}^{N+M-1}\) consists of two parts; (1) \(\left\{x_{i,k}\right\}_{k=0}^{N+L_{cl}-1}\), \(i\geq 0\), indicating the convolution between the information sequence \(\left\{s_{i,k}\right\}_{k=0}^{N-1}\) and the CIR; (2) \(\left\{d_{i,k}\right\}_{k=0}^{M+L_{cl}-1}=0\), \(i\geq 0\) indicating the linear convolution between the PN sequence \(\left\{c_{i,k}\right\}_{k=0}^{M-1}=0\), \(i\geq 0\), and the CIR. In this regard, the expressions of these parts are given by:  
\[x_{i,k}=s_{i,k}\oplus h_{i,k}=\sum_{l=0}^{L_{cl}-1}s_{i,k-1}\cdot h_{i,l},\;0 \leq k<N+L_{cl}-1 \tag{4}\]  
\[d_{i,k}=c_{i,k}\oplus h_{i,k}=\sum_{l=0}^{L_{cl}-1}c_{i,k-1}\cdot h_{i,l},\;0 \leq k<M+L_{cl}-1 \tag{5}\]  
where \(\oplus\) is the linear convolution operation.