The Delay-Doppler Signal Representation(時移多普勒信號表示)(3)

Channel Representation in Delay-Doppler

In communication, they are used to represent channels by means of a superposition of time and frequency shift operations.Figure 2, shows an example of the delay-Doppler representation of a specific channel which is composed of two main reflectors which share similar delay (range) but differ in their Doppler characteristic (velocities).
在通信中，我們通過時間和頻率變換操作的疊加來表示信道。下圖展示的是兩個有相似的時延，不同多普勒的主要反射體構成的特殊信道的時延多普勒信道表示。

Signal Representation in Delay-Doppler

The delay-Doppler signal representation is mathematically subtler and requires the introduction of a new class of functions called quasi-periodic functions. To this end, we choose a delay period $\tau_r$ and a Doppler period $\nu_r$ satisfying the condition $\tau_r\cdot\nu_r=1$, A delay-Doppler signal is a function $\phi(\tau,\nu)$ that satisfies the following quasi-periodicity condition$\phi(\tau+n\tau_r,\nu+m\nu_r)=e^{j2\pi(n\nu\tau_r-m\tau\nu_r)}\phi(\tau,\nu)$
爲了表示信號引入了一個準周期函數，滿足$\tau_r\cdot\nu_r=1$，這樣時延多普勒信號$\phi(\tau,\nu)$滿足$\phi(\tau+n\tau_r,\nu+m\nu_r)=e^{j2\pi(n\nu\tau_r-m\tau\nu_r)}\phi(\tau,\nu)$其中每次遍歷時延週期$\tau_r$得到相位因子$e^{j2\pi\nu\tau_r}$，對應的，每次遍歷多普勒週期$\nu_r$得到相位因子$e^{-j2\pi\tau\nu_r}$

Conversion among different representation

The conversion between the time and frequency representations is carried through the Fourier transform. The conversion between the delay-Doppler and the time and frequency representations is carried by the Zak transforms $Z_t$ and $Z_f$ respectively, The Zak
transforms are realized by means of periodic Fourier integration formulas
時間轉換到頻率是傅里葉變換，時延多普勒表示轉換成時間、頻率表示通過Zak變換，Zak變換通過週期傅里葉積分公式實現：$Z_t(\phi)=\int_0^{\nu_r}e^{j2\pi t\nu}\phi(t,\nu)d\nu$ $Z_f(\phi)=\int_0^{-\tau_r}e^{j2\pi t\nu}\phi(\tau,f)d\tau$準週期條件在二維到一維的變換過程中是很重要的，否則一個信號的時延多普勒表示是無限多的。
$Z_t(\phi)$是信號的時間表示？$Z_f(\phi)$是信號的頻率表示？
這個部分和之前看的paper的動不動就來的二重積分好像不一致。