Multicarrier Interpretation of OTFS（OTFS的多載波解釋）（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.

• 新定義的一個結果是，OTFS可以被看作是一個時-頻擴展方案，由一組在互反時-頻網格上定義的二維基函數(或碼字)組成。
• 另一個結果是，OTFS可以被設計成任意多載波調製(如OFDM)上的一個簡單預處理步驟
• 新的定義是基於延遲-多普勒平面上的網格和時頻平面上的互反網格之間的傅立葉對偶關係

2.Details(細節)

2.1Introduction of delay-doppler grid and time-frequency grid(時延多普勒網格和時間頻率網格的介紹)

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.

很有趣的一點是delay-doppler域是時延域有N點多普勒域有M點，時頻域是頻域有N點，時域有M點

2.2Fourier relation between the two grids(兩個網格之間的傅里葉關係)

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.

我們可以很簡單的發現SFFT變換等於對矩陣x(n,m)的行做n維FFT，對列做M維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.

因爲SFFT變換，我們在多載波收發器做預處理和後處理就可以實現OTFS收發機

2.3About symplectic exponential function(關於辛指數函數)

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.

多普勒座標決定這個函數關於時間的斜率，時延座標決定這個函數關於頻率的斜率

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.

從更廣闊的角度來看，傅里葉對偶關係delay-Doppler網格和時頻網格建立雷達和通信之間的數學聯繫,首先理論關心的是最大化的解決分離反射鏡/目標根據其delay-Doppler特點,第二個是關心最大化的信息量,可以可靠地通過這些反射鏡組成的通信通道。