The Delay-Doppler Signal Representation(时移多普勒信号表示)(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，我们通过时间和频率变换操作的叠加来表示信道。下图展示的是两个有相似的时延，不同多普勒的主要反射体构成的特殊信道的时延多普勒信道表示。

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 $\tau_r$τr​ and a Doppler period $\nu_r$νr​ satisfying the condition $\tau_r\cdot\nu_r=1$τr​⋅νr​=1, A delay-Doppler signal is a function $\phi(\tau,\nu)$ϕ(τ,ν) that satisfies the following quasi-periodicity condition$\phi(\tau+n\tau_r,\nu+m\nu_r)=e^{j2\pi(n\nu\tau_r-m\tau\nu_r)}\phi(\tau,\nu)$ϕ(τ+nτr​,ν+mνr​)=ej2π(nντr​−mτνr​)ϕ(τ,ν)
为了表示信号引入了一个准周期函数，满足$\tau_r\cdot\nu_r=1$τr​⋅νr​=1，这样时延多普勒信号$\phi(\tau,\nu)$ϕ(τ,ν)满足$\phi(\tau+n\tau_r,\nu+m\nu_r)=e^{j2\pi(n\nu\tau_r-m\tau\nu_r)}\phi(\tau,\nu)$ϕ(τ+nτr​,ν+mνr​)=ej2π(nντr​−mτνr​)ϕ(τ,ν)其中每次遍历时延周期$\tau_r$τr​得到相位因子$e^{j2\pi\nu\tau_r}$ej2πντr​，对应的，每次遍历多普勒周期$\nu_r$νr​得到相位因子$e^{-j2\pi\tau\nu_r}$e−j2πτνr​

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 $Z_t$Zt​ and $Z_f$Zf​ respectively, The Zak
transforms are realized by means of periodic Fourier integration formulas
时间转换到频率是傅里叶变换，时延多普勒表示转换成时间、频率表示通过Zak变换，Zak变换通过周期傅里叶积分公式实现：$Z_t(\phi)=\int_0^{\nu_r}e^{j2\pi t\nu}\phi(t,\nu)d\nu$Zt​(ϕ)=∫0νr​​ej2πtνϕ(t,ν)dν $Z_f(\phi)=\int_0^{-\tau_r}e^{j2\pi t\nu}\phi(\tau,f)d\tau$Zf​(ϕ)=∫0−τr​​ej2πtνϕ(τ,f)dτ准周期条件在二维到一维的变换过程中是很重要的，否则一个信号的时延多普勒表示是无限多的。
$Z_t(\phi)$Zt​(ϕ)是信号的时间表示？$Z_f(\phi)$Zf​(ϕ)是信号的频率表示？
这个部分和之前看的paper的动不动就来的二重积分好像不一致。