MIMO-OTFS in High-Doppler Fading Channels:Signal Detection and Channel Estimation(3)

3.3Vectorized formulation of the input-output relation

信道的脉冲响应可以写为
$h(\tau,\nu)=\sum_{i=1}^{P}h_i\delta(\tau-\tau_i)\delta(\nu-\nu_i)$即假设两根天线之间存在P条路径
??? Define $\tau_i=\frac{\alpha_i}{M\Delta f}$and $\nu_i=\frac{\beta_i}{NT}$ ,where αi and βi are integers denoting the indices of the delay tap (with delay $\tau_i$) and Doppler tap (with Doppler value $\nu_i$). In practice, although the delay and Doppler values are not exactly integer multiples of the taps, they can be well approximated by a few delay-Doppler taps in the discrete domain.
有了上面的假设之后，可以得到 $y[k,l]=\sum_{i=1}^{P}h_i'x[((k-\beta_i))_N,((l-\alpha_i))_M]+v[k,l]$用向量形式可以表示为 $y=Hx+v$ ??? $x_{k+Nl}=x[k,l]$，周期性？

4MIMO-OTFS Modulation

$y_{MIMO}=H_{MIMO}x_{MIMO}+v_{MIMO}$其中 $H_{MIMO}=\left[ \begin{matrix} H_{11} & H_{12} & \cdots & H_{1n_a} \\ H_{21} & H_{22} & \cdots & H_{2n_a} \\ \vdots & \vdots & \ddots & \vdots \\ H_{n_a1} & H_{n_a2} & \cdots & H_{n_an_a} \end{matrix} \right]$ $x_{MIMO}=[x_1^T,x_2^T,\cdots,x_{n_a}^T]^T,y_{MIMO}=[y_1^T,y_2^T,\cdots,y_{n_a}^T]^T,v_{MIMO}=[v_1^T,v_2^T,\cdots,v_{n_a}^T]^T$这个部分没有什么难理解的，和普通的MIMO很像。