# Superimposed Perfect Binary Array-Aided Channel Estimation for OTFS Systems
## 2 System Model  
We consider a set of \(MN\) modulated data symbols \(\{x_{d}[k,l],k\in[0,N-1],l\in[0,M-1]\}\) placed in DD-domain grids, where \(k\) and \(l\) represent Doppler shift and delay indices, respectively. The PBA symbol \(x_{p}[k,l]\), which will be introduced later, is superimposed on \(x_{d}[k,l]\) to yield the superimposed symbol \(x[k,l]\) in the DD domain for transmission, as follows:  
\[x[k,l]=x_{d}[k,l]+x_{p}[k,l],\;0\leq k\leq N-1,\;0\leq l\leq M-1. \tag{1}\]  
The \(MN\) PBA symbols in a single frame form a two-dimensional (2D) array \(\mathbf{P}\) that plays the role of pilot. The power levels of PBA and data symbols are assumed to be \(\rho_{p}=\mathbb{E}\Big{\{}\big{|}x_{p}[k,l]\big{|}^{2}\Big{\}}=\beta\rho\) and \(\rho_{d}=\mathbb{E}\Big{\{}\big{|}x_{d}[k,l]\big{|}^{2}\Big{\}}=(1-\beta)\rho\), respectively, where the zero-mean data symbols \(x_{d}[k,l]\) are independent and identically distributed (i.i.d. ), \(\rho\) is the power of each superimposed symbol, and \(\beta\in[0,1]\) denotes the power ratio between PBA and superimposed symbols. According to (1), the frame structure in SPBA-OTFS systems is illustrated in Figure 1b. The proposed method enables the whole OTFS frame to be used for data transmission when compared to the classic EP scheme from [13] shown in Figure 1a. The EP scheme embeds the single pilot symbol at \((k_{p},l_{p})\) and arranges the guard symbols between data and pilot symbols, i.e.,  
\[x[k,l]= \begin{cases}x_{p}[k,l]&k=k_{p},l=l_{p},\\ 0&k_{p}-2k_{max}\leq k\leq k_{p}+2k_{max},\\ &l_{p}-l_{max}\leq l\leq l_{p}+l_{max},\\ x_{d}[k,l]&\text{otherwise}.\end{cases}  \tag{2}\]  
where \(k_{max}\) and \(l_{max}\) denote the taps corresponding to the maximum Doppler and delay values, respectively. In the proposed method, the guard symbols are no longer needed as PBA symbols are superimposed on data symbols. Therefore, the SE is radically improved. Furthermore, Figure 2 presents a comparison of the transmitted data frames between the SP method from [17] and the proposed SPBA method. The SP method shown in Figure 2a requires an additional non-superimposed EP frame in Figure 1a to estimate the delay-Doppler taps for the following SP frames. In contrast, the proposed SPBA method can avoid the use of this dedicated pilot frame by utilizing the superimposed PBA to estimate delay-Doppler taps and complex path gains in the same SPBA frame, as shown in Figure 2b. Figure 1: The frame structure of (**a**) EP-OTFS systems and (**b**) SPBA-OTFS systems. Then, the \(MN\) superimposed symbols in the DD domain are mapped to the TF domain through an inverse symplectic finite Fourier transform (ISFFT), expressed as  
\[X[n,m]=\frac{1}{\sqrt{NM}}\sum_{k=0}^{N-1}\sum_{l=0}^{M-1}x[k,l]e^{j2\pi n\Delta f (t-nT)}, \tag{3}\]  
where \(n\in[0,N-1]\) and \(m\in[0,M-1]\) are time and subcarrier indices, respectively. Next, the values of \(X[n,m]\) are converted into a continuous-time waveform \(s(t)\) via Heisenberg transformation, i.e.,  
\[s(t)=\sum_{n=0}^{N-1}\sum_{m=0}^{M-1}X[n,m]g_{\mathrm{tx}}(t-nT)e^{j2\pi m\Delta f (t-nT)}, \tag{4}\]  
where \(T\) and \(\Delta f\) denote the symbol duration and subcarrier spacing, respectively, and \(g_{\mathrm{tx}}(t)\) is the transmit pulse. Doubly selective channels are stable and sparse in the DD domain, and the channel impulse response is given by  
\[h(\tau,v)=\sum_{i=1}^{P}h_{i}\delta(\tau-\tau_{i})\delta(v-v_{i}), \tag{5}\]  
where \(P\) is the number of propagation paths and \(h_{i}\) is the complex channel gain of the \(i\)-th path, distributed as \(\mathcal{CN}\Big{(}0,\sigma_{h_{i}}^{2}\Big{)}\)[13], which denotes the complex Gaussian random variable with zero mean and variance \(\sigma_{h_{i}}^{2}\). \(\tau_{i}\) and \(v_{i}\) denote the delay and Doppler values, respectively, of the \(i\)-th path. They can be expressed in terms of the delay index \(l_{i}\) and Doppler index \(k_{i}\) as \(\tau_{i}=l_{i}/M\Delta f\) and \(v_{i}=k_{i}/NT\). In this paper, only integer Doppler values are considered.