# Superimposed Perfect Binary Array-Aided Channel Estimation for OTFS Systems
## 3 Superimposed Perfect Binary Array-Aided OTFS Channel Estimator
### Design of the PBA  
According to Section 3.1, we wish to find a \(\mathbf{P}\) that satisfies (18) to expand the amplitude difference in \(\left|z\left[k_{j},l_{j}\right]\right|\) between the conditions where channels are successfully estimated or not. According to [19], a 2D PBA will meet this requirement. A matrix \(\mathbf{A}\in\mathbb{C}^{N\times M}\) is called a PBA if it satisfies the following conditions: (i) each element of \(\mathbf{A}\) is \(\pm 1\), and (ii) the periodic autocorrelation function (PACF) of \(\mathbf{A}\) is 0, which means that  
\[ \begin{split}\text{PACF}(u,v)&=\sum_{i=0}^{N-1}\sum_ {j=0}^{M-1}a[i,j]\cdot a[(i-u)_{N^{\prime}}(j-v)_{M}]\\ &=\text{sum}(\mathbf{A}\circ\mathbf{A}_{u,v})\\ &=\left\{\begin{array}{cc}NM,&(u,v)=(0,0)\\ 0,&(u,v)\
eq(0,0)\end{array}\right.\end{split} ,\end{split} \tag{24}\]  
where \(a[i,j]\) denotes the \((i,j)\)-th element of \(\mathbf{A}\). According to (24), if the superimposed array \(\mathbf{P}\) is a PBA, then (18) can be easily satisfied. Next, we will clarify how the PBA is created. Two kinds of 2D binary arrays, defined as follows, are used for the iterative construction of the PBA [20]. A matrix \(\mathbf{B}\in\mathbb{C}^{N\times M}\) is called a quasiperfect binary array (QPBA) if it satisfies the following conditions: (i) each element of \(\mathbf{B}\) is \(\pm 1\), and (ii) the periodic quasi-autocorrelation function (QACF) of \(\mathbf{B}\) is 0, which means that \[\text{QACF}(u,v) = \sum_{i=0}^{N-u-1}\sum_{j=0}^{M-1}b[i,j]\cdot b[(i-u)_{N^{\prime}}(j-v )_{M}] \tag{25}\] \[- \sum_{i=N-u}^{N-1}\sum_{j=0}^{M-1}b[i,j]\cdot b[(i-u)_{N^{\prime}} (j-v)_{M}]\] \[= \left\{ \begin{array}{cc}NM,&(u,v)=(0,0)\\ 0,&(u,v)\
eq(0,0)\end{array} \right.,\]  
where \(b[i,j]\) denotes the \((i,j)\)-th element of \(\mathbf{B}\). A matrix \(\mathbf{C}\in\mathbb{C}^{N\times M}\) is called a doubly quasiperfect binary array (DQPBA) if it satisfies the following conditions: (i) each element of \(\mathbf{C}\) is \(\pm 1\), and (ii) the periodic doubly quasi-autocorrelation function (DQACF) of \(\mathbf{C}\) is \(0\), which means that  
\[\text{DQACF}(u,v) = \sum_{i=0}^{N-u-1}\sum_{j=0}^{M-25s-1}c[i,j]\cdot c[(i-u)_{N^{ \prime}}(j-v)_{M}] \tag{26}\] \[- \sum_{i=N-u}^{N-1}\sum_{j=0}^{M-25s-1}c[i,j]\cdot c[(i-u)_{N^{ \prime}}(j-v)_{M}]\] \[- \sum_{i=0}^{N-u-1}\sum_{j=M-v}^{M-1}c[i,j]\cdot c[(i-u)_{N^{ \prime}}(j-v)_{M}]\] \[- \sum_{i=N-u}^{N-1}\sum_{j=M-v}^{M-1}c[i,j]\cdot c[(i-u)_{N^{ \prime}}(j-v)_{M}]\] \[= \left\{ \begin{array}{cc}NM,&(u,v)=(0,0)\\ 0,&(u,v)\
eq(0,0)\end{array} \right.,\]  
where \(c[i,j]\) denotes the \((i,j)\)-th element of \(\mathbf{C}\). There are four common ways to iteratively construct a high-order PBA, QPBA, and DQPBA using their low-order forms [21]. 1. Suppose that \(\mathbf{A}\in\mathbb{C}^{N\times M}\) and \(\mathbf{B}\in\mathbb{C}^{N\times M}\) are PBA and QPBA, respectively. A new PBA \(\mathbf{A}^{\prime}\in\mathbb{C}^{2N\times 2M}\) can be constructed as follows: \[a^{\prime}[i,j]=\left\{ \begin{array}{cc}a[(i)_{N^{\prime}},j/2],&(j)_{2}\equiv 0 \\ b[(i)_{N^{\prime}},(j-1)/2],&(j)_{2}\equiv 1,i<N\\ -b[(i)_{N^{\prime}},(j-1)/2],&(j)_{2}\equiv 1,i\geq N\end{array} \right.,\] (27) where \(a^{\prime}[i,j]\) denotes the \((i,j)\)-th element of \(\mathbf{A}^{\prime}\), \(i\in[0,2N-1]\) and \(j\in[0,2M-1]\). 2. Suppose that \(\mathbf{B}\in\mathbb{C}^{N\times M}\) and \(\mathbf{C}\in\mathbb{C}^{N\times M}\) are QPBA and DQPBA, respectively. A new QPBA \(\mathbf{B}^{\prime}\in\mathbb{C}^{2N\times 2M}\) can be constructed as follows: \[b^{\prime}[i,j]=\left\{ \begin{array}{cc}b[i/2,(j)_{M}],&(i)_{2}\equiv 0 \\ c[(i-1)/2,(j)_{M}],&(i)_{2}\equiv 1,j<M\\ -b[(i-1)/2,(j)_{M}],&(i)_{2}\equiv 1,j\geq M\end{array} \right.,\] (28) where \(b^{\prime}[i,j]\) denotes the \((i,j)\)-th element of \(\mathbf{B}^{\prime}\), \(i\in[0,2N-1]\) and \(j\in[0,2M-1]\). 3. Suppose that \(\mathbf{B}\in\mathbb{C}^{N\times M}\) is a QPBA and that \((M/\gcd(N,M))\bmod 2\equiv 1\), where the operator \(\gcd(N,M)\) outputs the greatest common divisor of \(N\) and \(M\).