Multicarrier Interpretation of OTFS（OTFS的多载波解释）（7）⭐

1.overview(概述)

In this section, we describe a variant of OTFS that is more adapted to the classical multicarrier formalism of time-frequency grids and filter-banks and illuminates aspects of OTFS that are not apparent from the Zak definition. One consequence of the new definition is that OTFS can be viewed as a time-frequency spreading scheme composed of a collection of two-dimensional basis-functions (or codewords) defined over a reciprocal time-frequency grid. Another consequence is that OTFS can be architected as a simple pre-processing step over an arbitrary multicarrier modulation such as OFDM. The new definition is based on Fourier duality relation between a grid in the delay-Doppler plane and a reciprocal grid in the time-frequency plane.

• 新定义的一个结果是，OTFS可以被看作是一个时-频扩展方案，由一组在互反时-频网格上定义的二维基函数(或码字)组成。
• 另一个结果是，OTFS可以被设计成任意多载波调制(如OFDM)上的一个简单预处理步骤
• 新的定义是基于延迟-多普勒平面上的网格和时频平面上的互反网格之间的傅立叶对偶关系

2.Details(细节)

2.1Introduction of delay-doppler grid and time-frequency grid(时延多普勒网格和时间频率网格的介绍)

The delay-Doppler grid consists of 𝑁 points along delay with spacing $\Delta\tau=\tau_r/N$ and 𝑀 points along Doppler with spacing $\Delta\nu = \nu_r/M$ and the reciprocal time-frequency grid consists of 𝑁 points along frequency with spacing $\Delta f = 1/\tau_r$ and 𝑀 points along time with spacing $\Delta t = 1/\nu_r$. The two grids are shown in Figure 12. The parameter 𝛥𝑡 is the multicarrier symbol duration and the parameter 𝛥𝑓 is the subcarrier spacing. The timefrequency grid can be interpreted as a sequence of 𝑀 multicarrier symbols each consisting of 𝑁 tones or subcarriers. We note that the bandwidth of the transmission 𝐵 =𝑀𝛥𝑓 is inversely proportional to the delay resolution 𝛥𝜏 and the duration of the
transmission 𝑇 = 𝑀𝛥𝑡 is inversely proportional to the Doppler resolution 𝛥𝜏.

很有趣的一点是delay-doppler域是时延域有N点多普勒域有M点，时频域是频域有N点，时域有M点

2.2Fourier relation between the two grids(两个网格之间的傅里叶关系)

The Fourier relation between the two grids is realized by a variant of the two-dimensional finite Fourier transform called the finite symplectic finite Fourier transform (SFFT for short). The SFFT sends an 𝑁×𝑀 delay-Doppler matrix 𝑥(𝑛Δ𝜏,𝑚Δ𝜐) to a reciprocal 𝑀×𝑁 time-frequency 𝑋(𝑚’Δ𝑡, 𝑛′Δ𝑓) via the following summation formula:$X(m'\Delta t,n'\Delta f)=\sum_{n=0}^{N-1}\sum_{m=0}^{M-1}e^{j2\pi(m'm/M-n'n/N)}x(n\Delta \tau,m\Delta \nu)$where the term “symplectic refers to the specific coupling form 𝑚′𝑚/𝑀 − 𝑛′𝑛/𝑁 inside the exponent. One can easily verify that the SFFT transform is equivalent to an application of an 𝑁-dimensional FFT along the columns of the matrix 𝑥(𝑛,𝑚) in conjunction with an 𝑀-dimensional IFFT along its rows.

我们可以很简单的发现SFFT变换等于对矩阵x(n,m)的行做n维FFT，对列做M维IFFT

The multicarrier interpretation of OTFS is the statement that the Zak transform of an 𝑁×𝑀 delay-Doppler matrix can be computed alternatively by first transforming the matrix to the time-frequency grid using the SFFT and then transforming the resulting reciprocal matrix to the time domain as a sequence of 𝑀 multicarrier symbols of size 𝑁 through a conventional multicarrier transmitter, i.e., IFFT transform of the columns.
Hence, using the SFFT transform, the OTFS transceiver can be overlaid as a pre- and post-processing step over a multicarrier transceiver. The multicarrier transceiver of OTFS is depicted in Figure 13 along with a visual representation of the doubly selective multiplicative CSC in the time-frequency domain and the corresponding invariant convolutive delay-Doppler CSC.

因为SFFT变换，我们在多载波收发器做预处理和后处理就可以实现OTFS收发机

2.3About symplectic exponential function(关于辛指数函数)

The multicarrier interpretation casts OTFS as a time-frequency spreading technique where each delay-Doppler QAM symbol 𝑥(𝑛Δ𝜏,𝑚Δ𝜐) is carried over a two-dimensional spreading ‘code’ or sequence on the time-frequency grid, given by the following symplectic exponential function:$\phi_{n,m}(m'\Delta t,n'\Delta f)=e^{j2\pi (mm'/M-nn'/N)}$ where the slope of this function along time is given by the Doppler coordinate 𝑚Δ𝜐 and the slope along frequency is given by the delay coordinate nΔτ (see examples in Figure 14). Thus, the analogy to two dimensional CDMA is seen, where the codewords are 2D complex exponentials that are orthogonal to each other.

多普勒座标决定这个函数关于时间的斜率，时延座标决定这个函数关于频率的斜率

3.Conclusion

From a broader perspective, the Fourier duality relation between the delay-Doppler grid and the time-frequency grid establishes a mathematical link between Radar and communication where the first theory is concerned with maximizing the resolution of separation between reflectors/targets according to their delay-Doppler characteristics and the second is concerned with maximizing the amount of information that can be reliably transmitted through the communication channel composed of these reflectors.

从更广阔的角度来看，傅里叶对偶关系delay-Doppler网格和时频网格建立雷达和通信之间的数学联系,首先理论关心的是最大化的解决分离反射镜/目标根据其delay-Doppler特点,第二个是关心最大化的信息量,可以可靠地通过这些反射镜组成的通信通道。